The Eulerian Functions

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Appendix A The Eulerian Functions

Here we consider the so-called Eulerian functions, namely the well-known Gamma function and Beta function together with some special functions that turn out to be related to them, such as the Psi function and the incomplete gamma functions. We recall not only the main properties and representations of these functions, but we also brieﬂy consider their applications in the evaluation of certain expressions relevant for the fractional calculus.

A.1 The Gamma Function

The Gamma function .z/ is the most widely used of all the special functions: it is usually discussed ﬁrst because it appears in almost every integral or series representation of other advanced mathematical functions. We take as its deﬁnition the integral formula Z 1 .z/ WD uz1 eu du ; Re z >0: (A.1.1) 0 This integral representation is the most common for , even if it is valid only in the right half-plane of C. The analytic continuation to the left half-plane can be done in different ways. As will be shown later, the domain of analyticity D of is

D D C nf0; 1; 2;:::;g : (A.1.2)

Using integration by parts, (A.1.1) shows that, at least for Re z >0, satisﬁes the simple difference equation

.z C 1/ D z .z/; (A.1.3)

© Springer-Verlag Berlin Heidelberg 2014 269 R. Gorenﬂo et al., Mittag-Lefﬂer Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2 270 A The Eulerian Functions which can be iterated to yield

.z C n/ D z .z C 1/::: .z C n 1/ .z/; n2 N : (A.1.4)

The recurrence formulas (A.1.3 and A.1.4) can be extended to any z 2 D .In particular, since .1/ D 1,wegetfornon-negative integer values

.n C 1/ D nŠ ; n D 0; 1; 2; : : : : (A.1.5)

As a consequence can be used to deﬁne the Complex Factorial Function

zŠ WD .z C 1/ : (A.1.6)

By the substitution u D v2 in (A.1.1)wegettheGaussian Integral Representation Z 1 2 .z/ D 2 ev v2z1dv ; Re .z/>0; (A.1.7) 0 which can be used to obtain when z assumes positive semi-integer values. Starting from Â Ã Z C1 1 2 p D ev dv D 1:77245 ; (A.1.8) 2 1 we obtain for n 2 N, Â Ã Z Â Ã C1 1 2 1 .2n 1/ŠŠ p .2n/Š n C D v vn vD D :1 e d n 2n (A.1.9) 2 1 2 2 2 nŠ

A.1.1 Analytic Continuation

A common way to derive the domain of analyticity (A.1.2) is to carry out the analytic continuation by the mixed representation due to Mittag-Lefﬂer: Z X1 .1/n 1 .z/ D C eu uz1 du ; z 2 D : (A.1.10) nŠ.z C n/ nD0 1

This representation can be obtained from the so-called Prym’s decomposition, namely by splitting the integral in (A.1.1) into two integrals, one over the interval

1The double factorial mŠŠ means mŠŠ D 1 3 :::mif m is odd, and mŠŠ D 2 4 :::mif m is even. A.1 The Gamma Function 271

0 Ä u Ä 1 which is then developed as a series, the other over the interval 1 Ä u Ä 1, which, being uniformly convergent inside C, provides an entire function. The terms of the series (uniformly convergent inside D ) provide the principal parts of at the corresponding poles zn Dn. So we recognize that is analytic in the entire complex plane except at the points zn Dn (n D 0; 1; : : : ), which turn n out to be simple poles with residues Rn D .1/ =nŠ. The point at inﬁnity, being an accumulation point of poles, is an essential non-isolated singularity. Thus is a transcendental meromorphic function. A formal way to obtain the domain of analyticity D is to carry out the required analytical continuation via the Recurrence Formula

.z C n/ .z/ D ; (A.1.11) .z C n 1/ .z C n 2/:::.z C 1/ z that is obtained by iterating (A.1.3) written as .z/ D .z C 1/=z.Inthiswaywe can enter the left half-plane step by step. The numerator on the R.H.S. of (A.1.11) is analytic for Re z > n; hence, the L.H.S. is analytic for Re z > n except for simple poles at z D 0; 1;:::;.n C 2/; .n C 1/.Sincen can be arbitrarily large, we deduce the properties discussed above. Another way to interpret the analytic continuation of the Gamma function is provided by the Cauchy–Saalschütz representation, which is obtained by iterated integration by parts in the basic representation (A.1.1). If n 0 denotes any non- negative integer, we have Z Ä 1 1 1 .z/D uz1 eu 1 C u u2 CC.1/nC1 un du (A.1.12) 0 2 n in the strip .n C 1/ < Re z < n. To prove this representation the starting point is provided by the integral Z 1 uz1 Œeu 1 du ; 1

Integration by parts gives (the integrated terms vanish at both limits) Z Z 1 1 1 1 uz1 Œeu 1 du D uzeu du D .z C 1/ D .z/: 0 z 0 z

So, by iteration, we get (A.1.12).

A.1.2 The Graph of the Gamma Function on the Real Axis

Plots of .x/ (continuous line) and 1=.x/ (dashed line) for 4

6 Γ(x)

2 1/Γ(x) 0

−2

−4

−6 −4 −3 −2 −1 0 1 2 3 4 x

Fig. A.1 Plots of .x/ (continuous line)and1=.x/ (dashed line)

2.5 Γ(x)

2 1/Γ(x) 1.5

X: 1.462 Y: 0.8856 0.5

0 0 0.5 1 1.5 2 2.5 3 3.5 4 x

Fig. A.2 Plots of .x/ (continuous line)and1=.x/ (dashed line)

Hereafter we provide some analytical arguments that support the plots on the real axis. In fact, one can get an idea of the graph of the Gamma function on the real axis using the formulas

.x/ .x C 1/ D x.x/; .x 1/ D ; x 1 to be iterated starting from the interval 0

For x>0the integral representation (A.1.1) yields .x/ > 0 and 00.x/ > 0 since Z Z 1 1 .x/ D eu ux1 du ;00.x/ D eu ux1 .log u/2 du : 0 0

As a consequence, on the positive real axis .x/ turns out to be positive and convex so that it ﬁrst decreases and then increases, exhibiting a minimum value. Since .1/ D .2/ D 1, we must have a minimum at some x0, 1

A.1.3 The Reﬂection or Complementary Formula .z/.1 z/ D : (A.1.13) sin z

This formula, which shows the relationship between the function and the trigonometric sin function, is of great importance together with the recurrence formula (A.1.3). It can be proved in several ways; the simplest proof consists in proving (A.1.13)for0

A.1.4 The Multiplication Formulas

Gauss proved the following Multiplication Formula

nY1 k .nz/ D .2/.1n/=2 nnz1=2 .z C /; nD 2;3;::: ; (A.1.14) n kD0 which reduces, for n D 2,toLegendre’s Duplication Formula

1 1 .2z/ D p 22z1=2 .z/.z C /; (A.1.15) 2 2 274 A The Eulerian Functions and, for n D 3,totheTriplication Formula

1 1 2 .3z/ D 33z1=2 .z/.z C /.z C /: (A.1.16) 2 3 3

A.1.5 Pochhammer’s Symbols

Pochhammer’s symbols .z/n are deﬁned for any non-negative integer n as

.z C n/ .z/ WD z .z C 1/ .z C 2/:::.z C n 1/ D ;n2 N ; (A.1.17) n .z/ with .z/0 D 1. In particular, for z D 1=2 ; we obtain from (A.1.9) Â Ã 1 .n C 1=2/ .2n 1/ŠŠ WD D : n 2 n .1=2/ 2

We extend the above notation to negative integers, deﬁning

.z C 1/ .z/ WD z .z 1/ .z 2/:::.z n C 1/ D ;n2 N : (A.1.18) n .z n C 1/

A.1.6 Hankel Integral Representations

In 1864 Hankel provided a complex integral representation of the function 1=.z/ valid for unrestricted z;itreads: Z 1 1 et .z/ D t; z 2 C ; z d (A.1.19a) 2i Hat where Ha denotes the Hankel path deﬁned as a contour that begins at t D1ia (a>0), encircles the branch cut that lies along the negative real axis, and ends up at t D1Cib (b>0). Of course, the branch cut is present when z is non-integer because t z is a multivalued function; in this case the contour can be chosen as in Fig. A.3 left, where C;above the cut; arg .t/ D ; below the cut:

When z is an integer, the contour can be taken to be simply a circle around the origin, described in the counterclockwise direction. A.1 The Gamma Function 275

Fig. A.3 The left Hankel contour Ha and the right Hankel contour HaC

An alternative representation is obtained assuming the branch cut along the positive real axis; in this case we get Z 1 1 et .z/ D t; z 2 C ; z d (A.1.19b) 2i HaC.t/ where HaC denotes the Hankel path deﬁned as a contour that begins at t DC1Cib (b>0), encircles the branch cut that lies along the positive real axis, and ends up at t DC1ia (a>0). When z is non-integer the contour can be chosen as in Fig. A.3 left, where 0; above the cut; arg .t/ D 2 ; below the cut:

When z is an integer, the contour can be taken to be simply a circle around the origin, described in the counterclockwise direction. We note that

i Ci Ha ! HaC if t ! t e ; and HaC ! Ha if t ! t e :

The advantage of the Hankel representations (A.1.19a)and(A.1.19b) compared with the integral representation (A.1.1) is that they converge for all complex z and not just for Rez >0. As a consequence 1= is a transcendental entire function (of maximum exponential type); the point at inﬁnity is an essential isolated singularity, which is an accumulation point of zeros (zn Dn; n D 0; 1; : : : ). Since 1= is entire, does not vanish in C. The formulas (A.1.19a)and(A.1.19b) are very useful for deriving integral representations in the complex plane for several special functions. Furthermore, using the reﬂection formula (A.1.13), we can get the integral representations of itself in terms of the Hankel paths (referred to as Hankel integral representations for ), which turn out to be valid in the whole domain of analyticity D . 276 A The Eulerian Functions

The required Hankel integral representations that provide the analytical continuation of turn out to be:

(a) Using the path Ha Z 1 t z1 .z/ D e t dt; z 2 D I (A.1.20a) 2i sin z Ha

(b) Using the path HaC Z 1 t z1 .z/ D e .t/ dt; z 2 D : (A.1.20b) 2i sin z HaC

A.1.7 Notable Integrals via the Gamma Function Z 1 .˛ C 1/ st t ˛ t D ; .s/ > 0 ; .˛/ > 1: e d ˛C1 Re Re (A.1.21) 0 s

This formula provides the Laplace transform of the power function t ˛ . Z 1 ˇ .1 C 1=ˇ/ at t D ; .a/>0; ˇ >0: e d 1=ˇ Re (A.1.22) 0 a p This integral for fixed a>0and ˇ D 2 attains the well-known value =a related to the Gauss integral.Forfixeda>0, the L.H.S. of (A.1.22) may be referred to as the generalized Gauss integral. The function I.ˇ/ WD .1 C 1=ˇ/ strongly decreases from infinity at ˇ D 0 to a positive minimum (less than the unity) attained around ˇ D 2 andthenslowly increases to the asymptotic value 1 as ˇ !1. The minimum value is attained at ˇ0 D 2:16638 : : : and I.ˇ0/ D 0:8856 : : : A more general formula is Z 1 1 .=/ 1 .1 C =/ zt t 1 t D D ; e d = = (A.1.23) 0 z z where Rez >0, >0,Re./ > 0. This formula includes (A.1.21)and(A.1.22); it reduces to (A.1.21)forz D s, D 1 and D ˛ C 1,andto(A.1.22)forz D a, D ˇ and D 1. A.1 The Gamma Function 277

A.1.8 Asymptotic Formulas Ä p 1 1 .z/ ' 2ez zz1=2 1 C C C ::: I (A.1.24) 12 z 288 z2 as z !1with jargzj <. This asymptotic expression is usually referred to as Stirling’s formula, originally given for nŠ. The accuracy of this formula is surprisingly good on the positive real axis and also for moderate values of z D x>0;as can be noted from the following exact formula, p xC xC1=2 xŠ D 2e. 12x / x I x>0; (A.1.25) where is a suitable number in .0; 1/. The two following asymptotic expressions provide a generalization of the Stirling formula. If a; b denote two positive constants, we have p .az C b/ ' 2eaz .az/azCb1=2 ; (A.1.26) as z !1with jargzj <,and Ä .z C a/ .a b/.a C b 1/ ' zab 1 C C ::: ; (A.1.27) .z C b/ 2z as z !1along any curve joining z D 0 and z D1provided z ¤a; a 1;:::, and z ¤b; b 1;::: :

A.1.9 Inﬁnite Products

An alternative approach to the Gamma function is via inﬁnite products, described by Euler in 1729 and Weierstrass in 1856. Let us start with the original formula given by Euler,

z 1 Y1 1 C 1 nŠ nz .z/ WD n D lim : (A.1.28) z 1 C z n!1 z.z C 1/:::.z C n/ nD1 n

The above limits exist for all z 2 D C : From Euler’s formula (A.1.28) it is possible to derive Weierstrass’ formula

1 h Á i 1 Y z D z e C z 1 C ez=n ; (A.1.29) .z/ n nD1 278 A The Eulerian Functions where C , called the Euler–Mascheroni constant,isgivenby 8 P < n 1 limn!1 kD1 k log n ; C D 0:5772157 : : : D : R (A.1.30) 0 1 u .1/ D 0 e log u du :

A.2 The Beta Function

A.2.1 Euler’s Integral Representation

The standard representation of the Beta function is Z 1 Re.p/ > 0 ; B.p; q/ D u p1 .1 u/ q1 du ; (A.2.1) 0 Re.q/ > 0 :

Note that, from a historical viewpoint, this representation is referred to as the Euler integral of the ﬁrst kind, while the integral representation (A.1.1)for is referred to as the Euler integral of the second kind. The Beta function is a complex function of two complex variables whose analyticity properties will be deduced later, as soon as the relation with the Gamma function has been established.

A.2.2 Symmetry

B.p; q/ D B.q; p/ : (A.2.2)

This property is a simple consequence of the deﬁnition (A.2.1).

A.2.3 Trigonometric Integral Representation Z =2 Re.p/ > 0 ; B.p; q/D2 .cos #/2p1 .sin #/2q1 d#; (A.2.3) 0 Re.q/ > 0 :

This noteworthy representation follows from (A.2.1) by setting u D .cos #/2. A.2 The Beta Function 279

A.2.4 Relation to the Gamma Function .p/.q/ B.p; q/ D : (A.2.4) .p C q/

This relation is of fundamental importance. Furthermore, it allows us to obtain the analytical continuation of the Beta function. The proof of (A.2.4) can easily be obtained by writing the product .p/.q/ as a double integral that is to be evaluated introducing polar coordinates. In this respect we must use the Gaussian representation (A.1.7) for the Gamma function and the trigonometric representation (A.2.3) for the Beta function. In fact, Z Z 1 1 2 2 .p/.q/ D 4 e.u Cv / u2p1 v2q1 du dv 0 0 Z Z 1 =2 2 D 4 e 2.pCq/1 d .cos #/2p1 .sin #/2q1 d# 0 0 D .p C q/B.p; q/ :

Henceforth, we shall exhibit other integral representations for B.p; q/, all valid for Rep>0;Req>0:

A.2.5 Other Integral Representations

Integral representations on Œ0; 1/ are 8 Z ˆ 1 xp1 ˆ dx; ˆ pCq ˆ 0 .1 C x/ ˆ <ˆ Z 1 xq1 B.p; q/ D dx; (A.2.5) ˆ pCq ˆ 0 .1 C x/ ˆ ˆ Z ˆ 1 p1 q1 :ˆ 1 x C x pCq dx: 2 0 .1 C x/

x The ﬁrst representation follows from (A.2.1) by setting u D 1Cx ; the other two are easily obtained by using the symmetry property of B.p; q/ : A further integral representation on Œ0; 1 is Z 1 yp1 C yq1 .p; q/ D y: B pCq d (A.2.6) 0 .1 C y/ 280 A The Eulerian Functions

This representation is obtained from the ﬁrst integral in (A.2.5)asasumofthetwo contributions Œ0; 1 and Œ1; 1/.

A.2.6 Notable Integrals via the Beta Function

The Beta function plays a fundamental role in the Laplace convolution of power functions. We recall that the Laplace convolution is the convolution between causal functions (i.e. vanishing for t<0), Z Z C1 t f.t/ g.t/ D f. /g.t /d D f. /g.t /d : 1 0

The convolution satisﬁes both the commutative and associative properties:

f.t/ g.t/ D g.t/ f.t/; f.t/ Œg.t/ h.t/ D Œf .t/ g.t/ h.t/ :

It is straightforward to prove the following identity, by setting in (A.2.1) u D =t ; Z t t p1 t q1 D p1 .t /q1 d D t pCq1 B.p; q/ : (A.2.7) 0

Introducing the causal Gel’fand–Shilov function

t 1 ˚ .t/ WD C ; 2 C ; . /

(where the sufﬁx C just denotes the causality property of vanishing for t<0), we can write the previous result in the following interesting form:

˚p.t/ ˚q.t/ D ˚pCq .t/ : (A.2.8)

In fact, dividing the L.H.S. of (A.2.7)by.p/.q/, and using (A.2.4), we obtain (A.2.8). As a consequence of (A.2.8) we get the semigroup property of the Riemann–Liouville fractional integral operator. In the following we describe other relevant applications of the Beta function. The results (A.2.7–A.2.8) show that the convolution integral between two (causal) functions, which are absolutely integrable in any interval Œ0; t and bounded in every ﬁnite interval that does not include the origin, is not necessarily continuous at t D 0, even if a theorem ensures that this integral turns out to be continuous for any t>0, see e.g. [Doe74, pp. 47–48]. In fact, considering two arbitrary real numbers ˛; ˇ greater than 1,wehave

˛ ˇ ˛CˇC1 I˛;ˇ.t/ WD t t D B.˛ C 1; ˇ C 1/ t ; (A.2.9) A.2 The Beta Function 281 so that 8 < C1 if 2<˛C ˇ<1; lim I˛;ˇ.t/ D c.˛/ if ˛ C ˇ D1; (A.2.10) ! C : t 0 0 if ˛ C ˇ>1; where c.˛/ D B.˛ C 1; ˛/ D .˛ C 1/ .˛/ D = sin.˛/. We note that in the case ˛ C ˇ D1 the convolution integral attains for any t>0the constant value c.˛/ :In particular, for ˛ D ˇ D1=2 ; we obtain the minimum value for c.˛/,i.e. Z t d p p D : (A.2.11) 0 t The Beta function is also used to prove some basic identities for the Gamma function, like the complementary formula (A.1.13)andtheduplication formula (A.1.15). For the complementary formula it is sufﬁcient to prove it for a real argument in the interval .0; 1/, namely .˛/.1 ˛/ D ;0<˛<1: sin ˛ We note from (A.2.4–A.2.5)that Z 1 x˛1 .˛/.1 ˛/ D B.˛; 1 ˛/ D dx: 0 1 C x

Then it remains to use well-known formula Z 1 x˛1 dx D : 0 1 C x sin ˛

To prove the duplication formula we note that it is equivalent to

.1=2/ .2z/ D 22z1 .z/.z C 1=2/ ; and hence, after simple manipulations, to

B.z;1=2/D 22z1 B.z; z/: (A.2.12)

This identity is easily veriﬁed for Re.z/>0, using the trigonometric representation (A.2.3) for the Beta function and noting that Z Z Z =2 =2 =2 .cos #/˛ d# D .sin #/˛ d# D 2˛ .cos #/˛ .sin #/˛ d#; 0 0 0 with Re.˛/ > 1,sincesin2# D 2 sin # cos #: 282 A The Eulerian Functions

A.3 Historical and Bibliographical Notes

For the historical development of the Gamma function we refer the reader to the notable article [Dav59]. It is surprising that the notation and the name Gamma function were ﬁrst used by Legendre in 1814 whereas in 1729 Euler had represented his function via an inﬁnite product, see Eq. (A.1.28). As a matter of fact Legendre introduced the representation (A.1.1) as a generalization of Euler’s integral expression for nŠ, Z 1 nŠ D . log t/n dt: 0

Indeed, changing the variable t ! u Dlog t,weget Z 1 nŠ D eu un du D .n C 1/ : 0

Lastly, we mention the well-known Bohr–Mollerup theorem which states that the -function is the only function which satisﬁes the relation f.z C 1/ D zf.z/ where log f.z/ is convex and f.1/ D 1. The proof of this fact is presented, e.g., in the book by Artin [Art64].

A.4 Exercises

A.4.1. The Pochhammer symbol is deﬁned as follows (see, e.g. [Tem96, p. 72])

.a C n/ .a/ WD a.a C 1/:::.aC n 1/ D : n .a/

Verify that for all a 2 C;m;nD 0; 1; : : : the following identities hold 8 <ˆ 0; if n>m; (a) .m/n D ˆ : n mŠ .1/ .mn/Š ; if n Ä mI n (b) .a/n D .1/ .a n C1/nI 2n a aC1 (c) .a/2n D 2 2 n 2 n I 2nC1 a aC1 (d) .a/2nC1 D 2 2 nC1 2 n : Prove the following formulas for suitable values of parameters A.4.2. ([Rai71, p. 103])

.1 C a=2/ cos a.1 a/ D 2 : .1C a/ .1 a=2/ A.4 Exercises 283

A.4.3. ([Rai71, p. 103])

.1 C a b/ sin .b a=2/.b a=2/ D : .1C a=2 b/ sin .b a/.b a/

A.4.4. Prove the formula ([Tem96,p.72]) Z 1 Á x 1 z t z1e˛t dt D ˛z=x; Re ˛>0;Re x>0;Re z >0: 0 x x

A.4.5. Verify the formulas ([Tem96, p. 73]) p (a) n C 1 D .1/n 22n nŠ ;nD 0; 1; 2; : : : : 2 .2n/Š 1 np 2n .2n/Š (b) n C 2 D .1/ 2 nŠ ;nD 0;1;2;:::: A.4.6. Verify the alternative reﬂection formula for the Gamma function ([Tem96, p. 74]) Â Ã Â Ã 1 1 1 z C z D ; z 62 Z; 2 2 cos z 2

in particular Â Ã Â Ã 1 1 iy C iy D ;y2 R: 2 2 cosh y

A.4.7. Verify the generalized reﬂection formula for the Gamma function ([Tem96, p. 74])

.z/.1 z/ .z n/ D .1/n .n C 1 z/ .1/n D ; z 62 Z;nD 0;1;2;:::: sin z.n C 1 z/

A.4.8. By using the reﬂection formula show that s 2 j.iy/j ejyj; as y !˙1: jyj

Calculate the following integrals A.4.9. ([Tem96, p. 74])

Z 2 .cos t/x cos ty dt; Re x>1: 0 284 A The Eulerian Functions

.x C 1/ Answer. h i . 2xC1 xCy xy 2 C 1 2 C 1

A.4.10. ([Tem96, pp. 76–77]) Z 1 t z1 log j1 tj dt; 1

cot z Answer. . z

A.4.11. ([Tem96, pp. 76–77], [Sne56]) R 1 z1 (a) 0 t cos t dt; Re z >1: z Answer. .z/ cos . R 2 1 z1 (b) 0 t sin t dt; Re z >1: z Answer. .z/ sin . 2 A.4.12.([WhiWat52, p. 300]) Show that for a<0, a D C ˛, 2 N;˛ >0, .x/.a/ X1 R D n C G .x/ ; .x C a/ x C n n nD1

where

.1/n.a 1/.a 2/:::.a n/ R C G.n/; n nŠ Á Á Á x x x G.x/ D 1 C 1 C ::: 1 C ; a 1 a 2 a G.x/ G.n/ G .x/ D : n x C n Appendix B The Basics of Entire Functions

B.1 Deﬁnition and Series Representations

A complex-valued function F W C ! C is called an entire function (or integral function) if it is analytic (C-differentiable) everywhere on the complex plane, i.e. if at each point z0 2 C the following limit exists

F.z/ F.z / lim 0 2 C: z!z0 z z0

Typical examples of entire functions are the polynomials, the exponential functions and also sums, products and compositions of these functions, thus trigonometric and hyperbolic functions. Among the special functions we point out the following entire functions: Airy functions Ai.z/; Bi.z/, Bessel functions of the first and second m;n kind J.z/; Y.z/,FoxH-functions Hp;q .z/ for certain values of parameters, the 1 reciprocal Gamma function .z/ , the generalized hypergeometric function pFq .z/, m;n Meijer’s G-functions Gp;q .z/, the Mittag-Leffler function E˛.z/ and its different generalizations, and the Wright function .zI ; ˇ/. According to Liouville’s theorem an entire function either has a singularity at infinity or it is a constant. Such a singularity can be either a pole (as is the case for a polynomial), or an essential singularity. In the latter case we speak of transcendental entire functions. All of the above-mentioned special functions are transcendental. Every entire function can be represented in the form of a power series

X1 k F.z/ D ck z ; (B.1.1) kD0 converging everywhere on C. Thus, according to the Cauchy–Hadamard formula, the coefﬁcients of the series (B.1.1) satisfy the following condition (the necessary

© Springer-Verlag Berlin Heidelberg 2014 285 R. Gorenflo et al., Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2 286 B The Basics of Entire Functions and sufficient condition for the sum of a power series to represent an entire function):

1 lim jckj k D 0: (B.1.2) k!1

The absolute value of the coefﬁcients of an entire function necessarily decreases to zero (although not monotonically, in general). One can classify the corresponding function in terms of the speed of this decrease (see below in Sect. B.2). Thus jckj! 0 for z !1is a necessary but not sufﬁcient condition for convergence of a power series.

B.2 Growth of Entire Functions: Order, Type and Indicator Function

The global behaviour of entire functions of finite order is characterized by their order and type. Recall (see e.g. [Lev56]) that the order of an entire function F.z/ is defined as an infimum of those k for which we have the inequality

rk MF .r/ WD max jF.z/j < e ; 8r>r.k/: jzjDr

Equivalently

log log M .r/ WD D lim sup F : (B.2.1) F r!1 log r

A more delicate characteristic of an entire function is its type. Recall (see e.g. [Lev56]) that the type of an entire function F.z/ of finite order is defined as the infimum of those A>0for which the inequality

Ar MF .r/ < e ; 8r>r.k/; holds. Equivalently

log M .r/ WD D lim sup F : (B.2.2) F r!1 r

For an entire function F.z/ represented in the form of the series

X1 k F.z/ D ckz kD0 B.3 Weierstrass Canonical Representation: Distribution of Zeros 287 its order and type can by found by the following formulae

klog k D lim supk!1 1 ; (B.2.3) log jc j k Á 1 1 p k . e/ D lim supk!1 k jckj : (B.2.4)

The asymptotic behaviour of an entire function is usually studied via its restriction to rays. In order to describe this we introduce the so-called indicator function of an entire function of order :

log jF.rei/j h./ D lim sup : (B.2.5) r!1 r For instance, the exponential function ez D expz has order D 1, type D 1 and indicator function h./ D cos ; Ä Ä .

B.3 Weierstrass Canonical Representation: Distribution of Zeros

The Weierstrass canonical representation generalizes the representation of complex polynomials in the form of a product of prime factors. By the fundamental theorem of algebra any complex polynomial of order n can be split into exactly n linear factors

Yn Pn.z/ D a .z ak/ with a 6D 0: (B.3.1) kD1

Entire functions can have infinitely many zeroes. In this case the finite product in (B.3.1) has to be replaced by an infinite product. Then the main question is to take an infinite product in such form that it can represent an entire function (so, is convergent on the whole complex plane). Another problem is that there are some entire functions which have very few zeroes (or have no zeroes at all, such as the exponential function). These two ideas were taken into account by Weierstrass. He took prime factors (or Weierstrass elementary factors) in the form 8 <ˆ .1 z/; if n D 0I En.z/ D ˆ n o (B.3.2) : z z2 zn .1 z/exp 1 C 2 C :::C n ; otherwise:

Usingsuchfactorsheproved 288 B The Basics of Entire Functions Theorem B.3.1 (Weierstrass). Let zj be a sequence of complex numbers j 2N0 (0 Djz0j < jz1jÄjz2jÄ:::), satisfying the following conditions:

(i) zj !1as j !1; (ii) There exists a sequence of positive integers pj j 2N such that ˇ ˇ X1 ˇ ˇ1Cpj ˇ z ˇ ˇ ˇ < 1: z j D1 j Then there exists an entire function which has zeroes only at the points zj , j 2N0 in particular, the following one Â Ã Y1 z F.z/ D zk E : (B.3.2a) pj z j D1 j

The following theorem is in a sense the converse of the above. Theorem B.3.2 (Weierstrass). Let F.z/ be an entire function and let zj be j 2N0 the zeros of F.z/, then there exists a sequence pj j 2N and an entire function g.z/ such that the following representation holds Â Ã Y1 z F.z/ D C zkeg.z/ E ; (B.3.3) pj z j D1 j where C is a constant and k 2 N0 is the multiplicity of the zero of F at the origin. For entire functions of finite order the Weierstrass theorems have a more exact form. Theorem B.3.3 (Hadamard). Let F.z/ be an entire function of finite order and let zj be the zeros of F.z/, listed with multiplicity, then the rank p of F.z/ is j 2N0 defined as the least positive integer such that ˇ ˇ X1 ˇ ˇ1Cp ˇ 1 ˇ ˇ ˇ < 1: (B.3.4) z j D1 j

Then the canonical Weierstrass product is given by Â Ã Y1 z F.z/ D C zkeg.z/ E ; (B.3.5) p z j D1 j B.4 Entire Functions of Completely Regular Growth 289 where g.z/ is a polynomial of degree q Ä . The genus of F.z/ deﬁned as max fp; qg is then also ﬁnite and

Ä : (B.3.6)

B.4 Entire Functions of Completely Regular Growth

Recall (see, e.g., [GoLeOs91, Lev56, Ron92]) that a ray arg z D is a ray of completely regular growth (CRG) for an entire function F of order if the following weak limit exists

log jF.rei/j lim ; (B.4.1) r!1 r where the term “weak limit” (lim in (B.4.1)) means that r tends to inﬁnity omitting the values of a set E RC which satisﬁes the condition

mes E \ .0; r/ lim D 0; r!1 r i.e. is relatively small. If all rays 2 Œ0; 2/ are rays of CRG for an entire function F (with the same exceptional set E D E ; 8), then such a function is called an entire function of completely regular growth. The main characterization of entire functions of completely regular growth is the following: An entire function F.z/ of order is a function of completely regular growth if and only if its set of zeroes zj has j 2N0 (a) In the case of a non-integer – an angular density:

n.r; ; / 4 . ; / WD lim I (B.4.1a) r!1 r (b) In the case of an integer – an angular density and a ﬁnite angular symmetry: 8 9 < = 1 X 1 a WD lim q C ; (B.4.1b) r!1 : ; zj 0

B.5 Historical and Bibliographical Notes

The period from 1850 to 1950 was the “golden century of the theory of entire functions”. This theory was one of the central subjects of complex analysis, it was rapidly developed in connection with several deep problems in mathematics as well as due to the usefulness of the analytic machinery for the solution of a wide range of applied problems. Here we mention only a few results of that time which have great influence on contemporary mathematics and which are related in some way to the subject of our book. Many of these results were influenced by Weierstrass (see, e.g., [Wei66], and the modern description of Weierstrass’s results in [AblFok97, Rud74], and [Kra04]). He introduced the notion of single-valued analytic functions (eindeutig analytis- che Funktionen), established the method of uniform convergence for families of complex functions which he successfully applied to the proof of his celebrated factorization theorem, introduced and applied the notion of (single-valued) analytic elements which became the core of the analytic continuation method, and developed the theory of elliptic functions illustrated by a special collection of these function now bearing his name and used by him and his successors to create and develop the analytic theory of complex differential equations. Mittag-Leffler was influenced by the genius of Weierstrass during his stay in Berlin (see Chap. 2 of this book). Among Mittag-Leffler’s results we point out the Mittag-Leffler factorization theorem for meromorphic functions which he obtained by developing the ideas of Weierstrass. The final form of this theorem was published in 1884 (see [Mit84]). The evolution of the hypotheses of the Mittag-Leffler theorem is worth studying; there were several varying publications of the theorem between 1876 and 1884, the majority of which were by Mittag-Leffler himself, and there is a noticeable evolution of the ideas which is marked by changes in notation from the original Weierstrassian style and the simplification of proofs. In particular, the later versions of Mittag-Leffler’s theorem differ markedly from those of his initial papers. Specifically, in 1882 we see a rather abrupt appearance of Cantor’s recently developed theory of transfinite sets in statements and proofs of Mittag-Leffler’s theorem, despite the generally negative reception of Cantor’s work by prominent mathematicians of that period. Mittag-Leffler was interested in the solution of the analytic continuation problem as applied to the study of the convergence of divergent series (we discuss the corresponding results in detail in Chap. 2). For this purpose he introduced a new entire function (now called the Mittag-Leffler function) which serves as the simplest generalization of the exponential function and also helped him to get a criterion for analytic continuation generalizing Borel’s result. As for the general theory of entire functions, we point out several results which constitute special directions in mathematics (see, e.g., [Boa54, Rud74]). First of all, from the results of Borel–Picard appeared the modern theory of value distributions for entire functions (see, e.g., [Lev56]) and Nevanlinna’s theory of value distributions for meromorphic functions (see, [Nev25, Nev36]). B.5 Historical and Bibliographical Notes 291

Levin and Pfluger discovered that the asymptotics of the distribution of zeros of entire functions and the growth of these functions at infinity are closely related for a large class of entire functions, namely the functions of completely regular growth. The theory of entire functions of completely regular growth of one variable, developed in the late 1930s independently by Levin and Pfluger, soon found applications in both mathematics and physics. Later, the theory was extended to functions in the half-plane, subharmonic functions in space, and entire functions of several variables. The monograph [Ron92] describes this theory and presents recent developments based on the concept of weak convergence. This enables a unified approach and provides a comparatively simple presentation of the classical Levin– Pfluger theory (see [GoLeOs91, Lev56]). This theory has also been generalized to analytic functions in angle domains (see [Gov94]). Rolf Nevanlinna’s most important mathematical achievement is the value distribution theory of meromorphic functions. The roots of the theory go back to the result of Picard in 1879, showing that a non-constant complex-valued function which is analytic in the entire complex plane assumes all complex values except at most one. In the early 1920s Rolf Nevanlinna, partly in collaboration with his brother Frithiof, extended the theory to cover meromorphic functions, i.e. functions analytic in the plane except for isolated points in which the Laurent series of the function has a finite number of terms with a negative power of the variable. Nevanlinna’s value distribution theory, or Nevanlinna theory, is crystallized in its two Main Theorems. Qualitatively, the first one states that if a value is assumed less frequently than average, then the function comes close to that value more often than average. The Second Main Theorem, more difficult than the first one, says that there are relatively few values which the function assumes less often than average. One more classical topic related to the subject of this book is the theory of complex differential equations. Many of the considered special functions (the Mittag-Leffler function and its generalizations are among them) satisfy certain complex differential equations. Thus, for the study of the properties of some functions it is natural to construct the corresponding differential equation and apply the general results of the theory. From another perspective, the Mittag-Leffler function and its generalizations serve as solutions of a new type of equation (in particular integral and differential equations of fractional order). The theory of the latter can be compared with the theory of classical complex differential equations. Among the results on complex differential equations we mention their relations to Nevanlinna Theory (see, e.g., [Lai93]) and those results which are related to the Riemann–Hilbert problem (see, [AnoBol94, Bol90]). In this appendix we collect a number of notions and results on entire functions since for certain values of parameters the Mittag-Leffler function and its generalization are entire functions. Thus we use in the main text approaches describing the asymptotics of entire functions, distribution of zeros, series and integral representations and other analytic properties. 292 B The Basics of Entire Functions

B.6 Exercises

Series. B.6.1. Represent the following functions in the form of Taylor series [Vo l 7 0, p. 65]. (a) cosh z, (b) sinh z, (c) sin2 z,(d)cosh2 z. Answers:

X1 z2n X1 z2nC1 .a/ I .b/ I .2n/Š .2n C 1/Š nD0 nD0

X1 22n1z2n 1 X1 22n1z2n .c/ .1/nC1 I .d/ C : .2n/Š 2 .2n/Š nD0 nD1

B.6.2. Prove the following formulas [Evg69, p. 134] (a)

X1 2. 2 22/ . 2 4n2/ cos. arcsin z/ D 1 .1/nz2nC2I .2n C 2/Š nD0

(b)

X1 . 2 12/ . 2 .2n 1/2/ sin. arcsin z/ D z .1/nz2nC1I .2n C 1/Š nD1 (c)

X1 22n1..n 1/Š/2 .arcsin z/2 D z2nI .2n/Š nD1 (d)

p 1 ln.z C 1 C z2/ X 22n..n/Š/2 p D .1/n z2nC1: 2 .2n C 1/Š 1 C z nD0

Inﬁnite products. B.6.3. Construct an inﬁnite product (Weierstrass product) for a function having the following collection of zeros [Mark66, p. 253]

(a) zk D k of multiplicity k; k 2 NI (b) zk D k of multiplicity j k j;k2 Z: B.6 Exercises 293

B.6.4. Prove the following formulas [LavSha65, p. 433] (a) Â Ã Y1 z2 sin z D z 1 I n22 nD1

(b) Â Ã z Y1 4z2 expz 1 D zexp 1 C I 2 4n22 nD1 (c) Â Ã Y1 4z2 cos z D 1 : .2n 1/22 nD1

B.6.5. Prove the following formulas [Evg69, pp. 266–267] (a) Â Ã z Y1 z2 cos D 1 I 2 .2n C 1/2 nD1 (b) Â Ã Y1 z z sin.z a/ D sin aexpz cot a 1 exp I a C n a C n nD1

(c)

1 Ä h i 2 Y z C a z a cos z cos a D .z2 a2/ 1 . /2 1 . /2 I 2 2n 2n nD1

(d)

1 h i z C a Y z a expz expa D exp .z a/ 1 C . /2 I 2 2n nD1

(e) Â Ã Y1 z4 cosh z cos z D z2 1 C I 44n4 nD1 294 B The Basics of Entire Functions

(f) Â Ã z2 Y1 z2 sin z z cos z D 1 I tan D >0I 3 2 n n nD1 n

(g) Â Ã sin z Y1 z z2 D 1 C I z.1 z/ n C n2 nD1

(h)

1 Á 1 Y z z D zexpC z 1 C exp ; .z/ n n nD1

where C is Euler’s constant; B.6.6. Prove the following formulas [GunKuz58, pp. 200–202] (a) Â Ã Y1 z2 sinh z D z 1 C I n22 nD1

(b) Â Ã Y1 .a b/2z2 expaz expbz D .a b/zexp1=2.a C b/z 1 C I 4n22 nD1

(d) Â Ã Y1 .z 1/4 expz2Cexp.2z 1/ D 2exp1=2.z2 C 2z 1/ 1 C : 2.2n 1/2 nD1 B.6 Exercises 295

B.6.7. Calculate the values of the following inﬁnite products [Vo l 7 0, p. 115] Y1 Â Ã Á 1 Q1 2 .a/ 1 D 1=2; .b/ n 4 D 1=4; n2 nD3 n21 nD2 Â Ã Y1 3 Á n 1 Q1 .c/ D 2=3; .d/ 1 C 1 D 2; n3 C 1 nD1 n.nC2/ nD2 Y1 Â Ã Á 2 Q1 nC1 .e/ 1 D 1=3; .f/ 1 C . 1/ D 1; n.n C 1/ nD1 n nD2 Á 2 2 2 Q p q 1 2n 2n .g/p p p D ;.h/ D C D : 2 p 2 n 1 2n 1 2n 1 2 2 C 2 2 C 2 C 2 B.6.8. Find domains of convergence for the following inﬁnite products [Vo l 7 0, pp. 116–117]

Â Ã Â Ã Y1 Y Y1 1 zn z2 .a/ .1 zn/; .b/ 1 C ;.c/ 1 ; nD1 2n n2 nD1 nD1 Â Ã Â Ã Y1 Y 1 1 1 2 .d/ 1 .1 /nzn ;.e/ 1 C .1 C /n zn ; n nD1 n nD1 1 Á Y z .f/ 1 C e.z=n/: n nD1

Answers:

.a/ j z j<1; .b/ j z j<1; .c/ j z j< 1; .d/ j z j>1; .e/ j z j<1=e;.f/ j z j< 1:

Characteristics of entire functions. B.6.9. Find the order and type of the following entire functions [Vo l 7 0, p. 122]

.a/ expazn;a>0;n2 NI .b/ znexp3zI .c/ z2exp2z exp3zI .d/ exp5z 3exp2z3I .e/ exp.2 i/z2I .f/ sin zI p .g/ cosh zI .h/ expz cos zI .i/ cos z: 296 B The Basics of Entire Functions

Answers:

.a/D n; D a.b/ D 1; D 3.c/ D 1; D 3 p .d/ D 3; D 2.e/ D 2; D 5.f/ D 1; D 1 p .g/ D 1; D 1.h/ D 1; D 2.i/D 1=2; D 1:

B.6.10. Calculate the order and type of the following entire functions, represented in the form of seriesP [Vo l 7 0, p. 124] P 1 z n 1 ln n n=a n (a) f.z/ D nD1 n ,(b)f.z/ D nD1 n z ;a >0, P P 1 1 n=a n 1 n2 n (c) f.z/ D nD2 n ln n z ;a >0,(d)f.z/ D nD0 e z , P P p 1 zn 1 cosh n n (e) f.z/ D D C ;a >0 (f) f.z/ D D z , n 1 nn1 a n 1 nŠ

1 X .1/nz2n .g/ zJ .z/ D ;.>1/ where J is the Bessel function. nŠ .n C C 1/ nD0

Answers:

.a/D 1; D 1=e .b/ D a; D1.c/ D 0; D 0 .d/ D 0; .e/D 0; .f/ D 1; D 1 .g/ D 1; D 2

B.6.11. Find the order and indicator function of the following entire functions [Vo l 7 0, p. 125] (a) expz,(b)exp.z/ C z2,(c)sinz z z .zn/ (d) cosp , (e) cosh , (f) exp , sin z p (g) z Answers: cos ; =2 Ä <=2 .a/D 1; h. / D cos .b/ D 1; h. / D 0; =2 Ä <3=2; .c/D 1; h. / Dj sin j;.d/ D 1; h. / Dj sin j; .e/D 1; h. / Dj cos j;.f/D n; h. / D cos n ; .g/ D 1=2; j sin =2 j :

Zeros of entire functions. B.6.12. Find all zeros and their multiplicities for the following entire functions [Vo l 7 0, p. 70] 2 3 .z22/2 sin z (a) .1 expz/.z 4/ ,(b)1 cos z,(c) z 3 sin3 z 3 (d) sin z,(e)z , (f) sin .z /, (g) cos3 z,(h)cosz3. B.6 Exercises 297

Answers: (a) z D˙2; 3rd orderI z D 2ki.k D 0; ˙1; / simple; (b) z D 2k.k D 0; ˙1; / 2nd order; (c) z D˙; 3rd orderI z D k.k D˙2;˙3; / simple; (d) z D k.k D 0; ˙1; / 3rd order; (e) z D p0; 2nd order; zpD k.k D˙p 1; ˙2;/ 3rd order; (f) z D 3 k;z D 1=2 3 k.1 ˙ i 3/; .k D˙1; ˙2;/ simple; (g) z D .2k C 1/ ;.k D 0; ˙1; / 3rd order; p 2 p p z D 3 .2k C 1/=2; z D 1=2 3 .2k C 1/=2.1 ˙ i 3/; (h) . .k D 0; ˙1; ˙2;/ simple B.6.13. Find the solutions of the following equations [Evg69,p.74] 4i 5 3i (a) sin z D 3 ,(b)sinz D 3 , (c) cos z D 4 , 3Ci i 1 (d) cos z D 4 ,(e)sinhz D 2 , (f) cosh z D 2 . Answers: (a) z D i.1/kln 3 C k;k D 0; ˙1; , (b) z D˙iln 3 C =2 C 2k; k D 0; ˙1; ; (c) z D˙.iln 2 C =4/ C 2k; k D 0; ˙1; , (d) z D˙.i=2ln 2 C =4/ C 2k; k D 0; ˙1; ; k i (e) z D .1/ 6 C k;k D 0; ˙1; ; i (f) z D˙3 C 2k; k D 0; ˙1; : Appendix C Integral Transforms

In this appendix we give an outline of the properties of some integral transforms. The main focus is on the properties which are often useful in treating applied problems. It is not our intention to present a complete theory of these transforms. In applications, however, it is advantageous to have at our disposal an arsenal of formal manipulations that should be used with a critical mind. Among the thousands of books on the subject we refer to a few in which the theory is developed with different degrees of rigour.

C.1 Fourier Type Transforms

The most general deﬁnition of the Fourier transform is

ZC1 .Ff /./ D F./ D A eiBtf.t/dt; A;B 2 R;A6D 0; B 6D 0: 1

In this book we use the following definition which is commonly used in Probability Theory and Stochastic Modelling. Definition C.1. The Fourier transform of a function f W R ! R.C/ is denoted by Ff D F./, 2 R, and defined by the integral

ZC1 .Ff /./ D F./ D eitf.t/dt; (C.1.1) 1

© Springer-Verlag Berlin Heidelberg 2014 299 R. Gorenflo et al., Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2 300 C Integral Transforms where F is called the Fourier transform operator or the Fourier transformation.1 The Fourier image Ff D F./is also denoted by f./O . From the theory of harmonic oscillations comes the following terminology: the pre-image (original) of the Fourier transform is usually called the signal or amplitude function depending on the time-variable t, while the Fourier image is called the spectral function depending on the frequency variable . The simplest class of functions for which the Fourier integral transform exists is the so-called class of rapidly decreasing functions, i.e. real- or complex-valued functions defined for all x 2 R and infinitely differentiable everywhere, such that each derivative tends to zero as jxj!C1faster that any positive power of x:

lim jxjN f .n/.x/ D 0 (C.1.2) jxj!C1 for each positive integer N and n. A more general sufﬁcient condition for a function f to have a Fourier transform is that f is absolutely integrable on R (see, e.g., [Doe74, p. 154]), i.e. belongs to 2 L1.R/ 8 9 < ZC1 = R R R C L1. / WD :f W ! . / W jf.x/jdx

Theorem C.1. Let the function f be absolutely integrable, i.e.

ZC1 jf.t/jdt<1: (C.1.3) 1

Then at every point t0,wheref is of bounded variation in some (arbitrarily small) neighbourhood of t0, the following inversion formula for the Fourier transform holds:

1Among other deﬁnitions of the Fourier transform we mention the symmetric form of the Fourier C1R transform .Ff /./ D p1 eitf.t/dt used in Functional Analysis, and .F'/.f / D 2 1 C1R ei.2f/t'.t/dt which is commonly used in Signal Processing. In the last deﬁnition the 1 variable t is time and f is the frequency of a signal. The function Ff is called the spectrum of the signal f.t/. 2 More precisely the space L1.R/ consists of equivalence classes with respect to the equivalence C1R relation: f g ” jf.x/ g.x/jdx D 0. 1 C.1 Fourier Type Transforms 301

ZC1 f.t C 0/ C f.t 0/ 1 0 0 D eitf./d;O (C.1.4) 2 2 1 where the integral is understood in the sense of the Cauchy principal value. If t0 is a point of continuity for f , then the value of the right-hand side of (C.1.4) coincides with the value f.t0/.

Moreover, if f 2 L1.R/ then its Fourier transform Ff D F./is a (uniformly) continuous and bounded function in 2 R, and lim F./ D 0. jj!C1 Theorem C.1 determines the inverse Fourier operator

ZC1 1 F 1F .t/ D eitf./d:O 2 1

In particular, if a function is locally integrable and has a compact support (i.e. vanishes outside some interval), then its Fourier transform exists. In this case, the Fourier transform F./ possesses an analytic continuation into the whole complex plane 2 C. In order to understand the meaning of the Fourier transform let us recall the Dirichlet condition. A real- or complex-valued function defined on the whole real line R is said to satisfy the Dirichlet condition on R if: (a) f.t/has in R no more than a finite number of finite discontinuity points (jump points) and has no infinite discontinuity points; (b) f.t/has in R no more than a finite number of maximum and minimum points. If f.t/satisfies the Dirichlet condition in R and is absolutely integrable, then the following Fourier integral formula holds:

ZC1 ZC1 f.t C 0/ C f.t 0/ 1 D eitd f./eid (C.1.5) 2 2 1 1 at any ﬁnite discontinuity point t 2 .1; C1/. This result is also known as the Fourier integral theorem. In particular, if f is continuous at t, then the Fourier integral formula can be written as

ZC1 ZC1 1 f.t/ D eitd f./eid: (C.1.6) 2 1 1 302 C Integral Transforms

Let us recall some basic properties of the Fourier transform (for more detailed information we refer to the treatise by Titchmarsh (see [Tit86])). Á Ff ./ D f.O /: (C.1.7)

Z1 If f.t/ D f.t/; then .Ff/./D 2 f.t/cos t dt: (C.1.8)

0 Z1 If f.t/ Df.t/; then .Ff/./D2i f.t/sin t dt: (C.1.9)

0 Ff.a1t C b/ ./ D aeiabf.a/;a>0;O (C.1.10) Â Ã 1 b Feibtf.at/ ./ D fO ;a>0; (C.1.11) a a d nf./O .Ft nf.t//./D i n ;n2 N; (C.1.12) dn Ff .n/.t/ ./ D i nnf./;nO 2 N; (C.1.13)

.F.f g/.t// ./ D f./O Og./; (C.1.14) provided that all Fourier images on the left- and right-hand sides exist, where .f g/ .t/ is the Fourier convolution,i.e.

ZC1 .f g/.t/ D f.t /g. /d : (C.1.15) 1

Sufﬁcient conditions for the fulﬁllment of equality (C.1.14) read that this equality holds if both functions f and g are integrable and square integrable on the real line: \ f; g 2 L1.R/ L2.R/: (C.1.16)

These conditions coincide with the conditions which guarantee the fulﬁllment of the Cauchy–Schwarz inequality:

ZC1 ZC1 ZC1 jf.t/ g.t/jdt Ä jf.t/j2dt jg.t/j2dt: (C.1.17) 1 1 1

Analogous conditions give us the so-called Parseval identity for the Fourier transform. C.1 Fourier Type Transforms 303

Theorem C.2. Let f be integrable and square integrable on the real line, i.e. \ f 2 L1.R/ L2.R/:

Then for all real x the following formula holds:

ZC1 ZC1 1 f.t/f.t x/dt D eixf./O f./O d: (C.1.18) 2 1 1

Moreover for x D 0 this yields the Parseval formula

ZC1 ZC1 1 jf.t/j2dt D jf./O j2d: (C.1.19) 2 1 1

The Mittag-Lefﬂer function is one of the most important special functions related to the Fractional Calculus. Thus we recall two composition formulas for the Fourier transform and fractional integrals/derivatives.

'./O FI ˛ ' ./ D ;0

˛ where I˙ are fractional integrals of the Liouville type:

Zx ZC1 1 '.t/dt 1 '.t/dt I ˛ ' .x/ D ; I ˛' .x/ D : C .˛/ .x t/1˛ .˛/ .t x/1˛ 1 x

Formulas (C.1.20) are valid for any function ' 2 L1.1; C1/. Corresponding formulas for fractional derivatives of the Liouville type

Zx 1 dn '.t/dt D˛ ' .x/ D ; C .n ˛/ dxn .x t/˛nC1 1 ZC1 .1/n dn '.t/dt D˛ ' .x/ D ;nD ŒRe ˛ C 1; .n ˛/ dxn .t x/1˛ x have the form FD˛ ˛ ˙' ./ D . i/ './;O Re ˛>0: (C.1.21) 304 C Integral Transforms

Formulas (C.1.21) are valid, in particular, for all functions having derivatives up to the order n D ŒRe ˛ C 1, and rapidly decreasing at inﬁnity together with all derivatives (see, e.g. [SaKiMa93]). In (C.1.20)and(C.1.21) the values of the function . i/˛ are calculated according to the formula

˛ ˛ log jxj ˛i sgn x . ix/ D e 2 :

We mention two formulas for cosine- and sine-integral transforms of the Riemann–Liouville fractional integral (see, e.g., [SaKiMa93, p. 140]) h i ˛ ˛ F I ˛ ' ./ D ˛ cos .F '/./ sin .F '/./ ;>0; (C.1.22) c 0C 2 c 2 s h i ˛ ˛ F I ˛ ' ./ D ˛ sin .F '/./ C cos .F '/./ ;>0: (C.1.23) s 0C 2 c 2 s The above results are considered in the classical case and in the distributional sense (see, e.g., [Bre65]). We use them in the text only in the ﬁrst sense.

C.2 The Laplace Transform

The classical Laplace transform is deﬁned by the following integral formula

Z1 .Lf/.s/D estf.t/dt; (C.2.1)

0 provided that the function f (the Laplace original) is absolutely integrable on the semi-axis .0; C1/. In this case the image of the Laplace transform (also called the Laplace image), i.e. the function

F.s/ D .Lf/.s/ (C.2.2)

(sometimes denoted F.s/ D f.s/Q ) is deﬁned and analytic in the half-plane Re s>0. It may happen that the Laplace image can be analytically continued to the left of the imaginary axes Re s D 0 in a bigger domain, i.e. there exist a non-positive real number s (called the Laplace abscissa of convergence) such that F.s/ D f.s/Q is analytic in the half-plane Re s s . Then the following inverse Laplace transform can be introduced Z 1 L1F .t/ D estF.s/ds; (C.2.3) 2i Lic C.2 The Laplace Transform 305 where Lic D .c i1;cC i1/, c> s , and the integral is usually understood in the sense of the Cauchy principal value, i.e.

Z cZCiT estF.s/ds D lim estF.s/ds: (C.2.4) T !C1 Lic ciT

If the integral (C.2.2) converges absolutely on the line Re s D c,thenatany continuity point t0 of the original f the integral (C.2.3) gives the value of f at this point, i.e. Z 1 est0 f.s/Q ds D f.t /: (C.2.5) 2i 0 Lic

Thus under these conditions operators L and L1 constitute an inverse pair of operators. Correspondingly, the functions f and F D fQ constitute a Laplace transform pair. To describe this fact the following notation is used Z 1 st f.t/ f.s/Q D e f.t/dt; Re s> c ; (C.2.6) 0 where c is the abscissa of the convergence. Here the sign denotes the juxtaposition of a function (depending on t 2 RC) with its Laplace transform (depending on s 2 C). In the following the conjugate variables ft;sg may be different, e.g., fr; sg and the abscissa of the convergence may sometimes be omitted. Furthermore, throughout our analysis, we assume that the Laplace transforms obtained by our formal manipulations are invertible by using the standard Bromwich formula. Among the rules for the Laplace transform pairs we recall the following one, which turns out to be useful for our purposes, Z 1 1=2 1 2 f.sQ / p r =.4t/ f.r/ r : e d 1=2 (C.2.7) t 0 s

Since an examination of convergence conditions is not always possible (see, e.g., [Wid46]) sometimes the terminology “Laplace transform pair” is used for pairs f; fQ not necessarily satisfying the equality (C.2.5) at certain points. There are several properties of the Laplace transform which make it very useful in the study of a wide class of differential and integral equations. Let us recall a few main properties of the Laplace transform (in the form of Laplace transform pairs) (more information can be found, e.g., [BatErd54a, Wid46, Doe74]).

eatf.t/ f.sQ C a/; a > 0I (C.2.8) 306 C Integral Transforms

dnfQ t nf.t/ .1/n .s/; n 2 NI (C.2.9) dt n Z1 Z1 Z1 n t f.t/ dsn dsn1 ::: f.sQ 1/ds1;n2 NI (C.2.10)

s sn s2 f .n/.t/ snf.s/Q sn1f.0/ sn2f 0.0/ ::: f .n1/.0/; n 2 NI (C.2.11)

Zt Ztn Zt2 n dtn dtn1 ::: f.t1/dt1 s f.s/;nQ 2 NI (C.2.12) 0 0 0 Â Ã Â Ã d n d n t f.t/ s f.s/;nQ 2 NI (C.2.13) dt ds Â Ã Â Ã d n d n t f.t/ s f.s/;nQ 2 NI (C.2.14) dt ds

.f1 f2/.t/ fQ1.s/ fQ2.s/; (C.2.15) where

.f1 f2/.t/D f1. /f2.t /d (C.2.16) 0 is the so-called Laplace convolution. Among simple sufﬁcient existence conditions for the Laplace transform we point out the following: The Laplace transform exists in the half-plane Re s>a(a>0) provided that the original f is locally integrable on RC D .0; C1/ and has an exponential growth of order a at inﬁnity, i.e. there exists a positive constant K>0and positive t0 >0 such that

at jf.t/j

From the formula (C.2.15) follows immediately the Laplace transform of the Riemann–Liouville fractional integral L ˛ ˛ L I0C' .s/ D s . '/ .s/; Re ˛>0: (C.2.17)

The Laplace transform of the Riemann–Liouville fractional derivative is given by the formula (see, e.g., [OldSpa74, p. 134]) ˇ Xn ˇ L ˛ ˛ L ˛k ˛ ˇ D0C' .s/ D s . '/.s/ s D0C' .t/ ;n 1

Xn1 L C ˛ ˛ L ˛k1 D0C' .s/ D s . '/.s/ s '.0C/; n 1

˛ Formula (C.2.19) is simpliﬁed in the case of the Marchaud fractional derivative DC ˛ and Grünwald–Letnikov fractional derivative GLD0C: L ˛ ˛ L L ˛ ˛ L DC' .s/ D s . '/ .s/; GLD0C' .s/ D s . '/ .s/: (C.2.20)

C.3 The Mellin Transform

Let Z C1 s1 M ff.r/I sgDf .s/ D f.r/r dr; 1 < Re .s/ < 2 (C.3.1) 0 be the Mellin transform of a sufﬁciently well-behaved function f.r/,andlet Z 1 Ci1 M1 ff .s/I rgDf.r/D f .s/ rs ds (C.3.2) 2i i1 be the inverse Mellin transform, where r>0, D Re .s/, 1 <<2. For the existence of the Mellin transform and the validity of the inversion formula we need to recall the following theorems, adapted from Marichev’s treatise [Mari83], see THEOREMS 11, 12, on page 39. Theorem C.3. Let f.r/ 2 Lc.; E/ ; 0 < < E < 1, be continuous in the intervals .0; ; ŒE; 1/, and let jf.r/jÄMr1 for 0E,whereM is a constant. Then for the existence of a strip in the s-plane s1 c in which f.r/r belongs to L .0; 1/ it is sufﬁcient that 1 <2 : When this condition holds, the Mellin transform f .s/ exists and is analytic in the vertical strip 1 <D Re.s/ < 2 : Theorem C.4. If f.t/is piecewise differentiable, and f.r/r1 2 Lc .0; 1/,then the formula (C.3.2) holds true at all points where f.r/ is continuous and the (complex) integral in it must be understood in the sense of the Cauchy principal value. We refer to specialized treatises and/or handbooks, see, e.g. [BatErd54a,Mari83, PrBrMa-V3], for more details and tables on the Mellin transform. Here, for our convenience we recall the main rules that are useful when adapting the formulae from the handbooks and which are also relevant in the following. 308 C Integral Transforms

M Denoting by $ the juxtaposition of a function f.r/ with its Mellin transform f .s/, the main rules are:

M f.ar/ $ as f .s/ ; a > 0 ; (C.3.3)

M ra f.r/ $ f .s C a/ ; (C.3.4)

M 1 f.rp / $ f .s=p/ ; p ¤ 0; (C.3.5) jpj Z1 1 M h.r/ D f ./ g.r=/ d $ h.s/ D f .s/ g.s/ : (C.3.6) 0

The Mellin convolution formula (C.3.6) is useful in treating integrals of Fourier type for x Djxj >0: Z 1 1 I .x/ D f./ cos . x/ d; (C.3.7) c Z0 1 1 Is.x/ D f./ sin . x/ d; (C.3.8) 0 when the Mellin transform f .s/ of f./ is known. In fact we recognize that the integrals Ic.x/ and Is.x/ can be interpreted as Mellin convolutions (C.3.6) between f./and the functions gc ./ ; gs./, respectively, with r D 1=jxj ;D ;where Â Ã Á 1 1 M .1 s/ s g ./ WD cos $ sin WD g.s/; 0 < Re.s/ < 1 ; c jxj jxj 2 c (C.3.9) Â Ã Á 1 1 M .1 s/ s g ./ WD sin $ cos WD g.s/; 0 < Re.s/ < 2 : s jxj jxj 2 s (C.3.10)

The Mellin transform pairs (C.3.9)and(C.3.10) have been adapted from the tables in [Mari83]byusing(C.3.3)–(C.3.5) and the duplication and reﬂection formulae for the Gamma function. Finally, the inverse Mellin transform representation (C.3.2) provides the required integrals as Z Ci1 Á 1 1 s s Ic.x/ D f .s/ .1 s/ sin x ds; x >0; 0< <1; x 2i i1 2 (C.3.11) C.3 The Mellin Transform 309 Z Ci1 Á 1 1 s s Is.x/ D f .s/ .1 s/ cos x ds; x >0; 0< <2: x 2i i1 2 (C.3.12)

First, we present some known properties of the Mellin integral transform.

k 1 Theorem C.5. Let x 2 f.x/ 2 L2.0; C1/. Then the following four statements hold: 10. The functions Z a M.sI a/ D f.x/xs1dx; Re s D k; (C.3.13) 1=a converge in the mean when a !C1on the line .k i1;kC i1/, that is, there exists a function M.s/ 2 L2.k i1;kC i1/ such that Z kCi1 lim jM.s/ M.sI a/j2 jdsjD0: (C.3.14) a!C1 ki1

20. The functions Z 1 kCia f.x;a/ D M.s/xs ds; 0 < x < C1 (C.3.15) 2i kia converge in the mean on the semi-axis with the weight function x2k1 to the function f.x/when a !C1, that is, Z C1 lim jf.x/ f.x;a/j2x2k1 dx D 0; (C.3.16) a!C1 0 moreover, almost everywhere on the semi-axis .0; C1/ we have Z 1 d kCi1 x1s f.x/ D M.s/ ds: (C.3.17) 2i dx ki1 1 s

30. The Parseval identity Z Z C1 1 C1 jf.x/j2x2k1dx D jM.k C it/j2 dt (C.3.18) 0 2 1 is valid. 0 4 . Conversely, for any function M.s/ 2 L2.ki1;kCi1/ the functions (C.3.15) converge in the mean when a !C1in the sense of (C.3.16) to some function f.x/ 2 L.0; C1/ which can be represented in the form (C.3.17). The functions 310 C Integral Transforms

(C.3.13) converge in the mean in the sense of (C.3.14) to the function M.s/ when a !C1and, moreover, the Parseval identity (C.3.18) is valid. We present a number of important formulas for the Mellin transform.

.M x˛f.x//.p/D f .p C ˛/; (C.3.19) Â Ã .1 x/˛1H.1 x/ .p/ M .p/ D ; (C.3.20) .˛/ .pC ˛/ where H.x/ is the Heaviside function 1; 0 Ä x

Z1 1 W ˛f.x/ D .t x/˛1f.t/dt; 0 < Re ˛ < 1; x > 0: (C.3.22) .˛/ x

The Mellin transform of the Riemann–Liouville fractional integral is given by the formula

.1 ˛ p/ M I ˛ f.x/ .p/ D f .p C ˛/; Re .˛ C p/ < 1: (C.3.23) 0C .1 p/

The Mellin transform of the Riemann–Liouville fractional derivative is represented in the form

.1C ˛ p/ M D˛ f.x/ .p/ D f .p ˛/ 0C .1 p/

Xn1 .1C k p/ xD1 C D˛k1f .x/xpk1 ; (C.3.24) .1 p/ 0C xD0 kD0 and the formula for the Mellin transform of the Caputo derivative has the form

.1C ˛ p/ M CD˛ f.x/ .p/ D f .p ˛/: (C.3.25) 0C .1 p/ C.4 Simple Examples and Tables of Transforms of Basic Elementary and. . . 311

C.4 Simple Examples and Tables of Transforms of Basic Elementary and Special Functions

Table of selected values of the Fourier integral transform.

.Ff /./ D F./ D f./O C1R n/n f(t) D eitf.t/dt Conditions 1 2a 1. exp.ajtj/ .a2 C2/ a>0 4ai 2. texp.ajtj/ 2 C 2 2 a>0 p.a / Á 2 2 3. exp.at / a exp 4a a>0 1 4. t2Ca2 a exp.ajj/ a>0 t i 5. (t2Ca2 2a exp.ajj/ a>0 c; a Ä t Ä b; ˚ « 6. ic eib eia a 0 0; otherwise sin at 7. t 2H .a jj/ a>0 C C ˛ cos . 2 .˛ 1// ˛ 2 ;˛ 6D 0; 8. jtj 2 .˛ C 1/ jj1C˛ ˛ 6D1 2k; k 2 N0 cos . ˛ / ˛ 2 N 9. jtj sgn t 2i .˛ C 1/ jj1C˛ sgn ˛ 6D2k; k 2 i 10. expft.a i!/gH.t/ !CCia a>0 j j H.ap t / 11. 2J0.a/ a>0 a2t2 aCi 12. exp.at/H.t/ a2Ch2 i a>0 1 .i/n 1 N 13. tn i .n1/Š sign n 2 2ı./" # ˇ ˇ 2 .2; 2/; .1; 1/ ˇ 1 14. E˛.jtj/ 2 21 2 ˛>1 .˛ C 1;" 2˛/ # ˇ .2; 2/; .1; 1/ ˇ 15. E .jtj/ 2ı./ 2 ˇ 1 ˛>1;ˇ2 C ˛;ˇ .ˇ/ 2 2 1 .˛ C ˇ; 2˛/ 2

Table of selected values of the Laplace integral transform.

C1R n/n f.t/ .Lf /.s/ D F.s/ D f.s/Q D estf.t/dt Conditions 0 a .aC1/ 1. t saC1 a>1 at sa 2. e cos bt .sa/2 Cb2 ˛1 1 ˛ s Re ˛ E˛.at / s˛ a Re ˛>0;s>jaj at b 3. e sin bt .sa/2 Cb2 s2b2 4. t cos bt .s2Cb2/2 (continued) 312 C Integral Transforms

.Lf /.s/ D F.s/ C1R n/n f.t/ D f.s/Q D estf.t/dt Conditions 0 5. t sin bt 2bs p.s2Cb2/2 p 6. p1 exp a exp 2 as t t s C C Z ˛1 .˛/ 1 a; b 2 ;c 2 n 0 ; 7. t 2F1.a; bI cI t/ ˛ 3F1.˛;a;bI cI / s s Re ˛>0I Re s>0 C C Z ˛1 .˛/ 1 a 2 ;c 2 n 0 ; 8. t ˚.aI cI t/ ˛ 2F1.˛; aI bI / s s Re ˛>0I Re s>0 .˛/ .˛c1/ C C Z .acC˛C1/ a 2 ;c 2 n 0 ; ˛1 9. t .aI cI t/ 2F1.˛; ˛ c C 1I Re ˛>0I Re c<1C Re ˛I a c C ˛ C 1I 1 1 / j1 sj <1 p s 10. exp a2t erf a t p a p s.sa2/ 11. exp a2t erfc a t p pa s. sCa/ ˛1 1 ˛ s Re ˛ 12. E˛.at / s˛ a Re ˛>0;s>jaj ˛1 1 ˛ s Re ˛ 13. E˛.at / s˛ a Re ˛>0;s>jaj

ˇ1 ˛ s˛ ˇ Re ˛>0;Re ˇ>0; 14. t E˛;ˇ.at / s˛ a 1 s>jaj Re ˛ s

m˛Cˇ1 .m/ mŠs˛ ˇ Re ˛>0;Re ˇ>0; 15. t E˛;ˇ .at/ .s˛ a/mC1 1 m 2 N;s >jaj Re ˛ " # ˇ Re ˛>0;Re ˇ>0; ˇ .1; 1/; .; / 1 1 a ˇ 16. t E˛;ˇ.at / s 21 s˛ Re >0;Re >0; .ˇ; ˛/ 1 s>jaj Re ˛ Re ˛>0;Re ˇ>0; ˇ1 ˛ ˇ ˛ 17. t E˛;ˇ.at / s .1 as / Re >0; 1 s>jaj Re ˛ " # ˇ Re ˛>0;Re ˇ>0; ˇ .ı; 1/; .1; 1/ ı 1 1 ˇ 18. Eˇ; .t/ s 21 s Re >0;Re ı>0; .ˇ; / 1 " # Re >0;s>jaj Re ˛ ˇ ˇ 1 1 ˇ .; 1/; .1; 1/ Re ˛j >0;Re ˇj >0; 19. E..˛j ;ˇj /1;mI t/ s 2m s 1 .˛j ;ˇj /1;m Re >0;s>jaj Re ˛ C.5 Historical and Bibliographical Notes 313

Table of selected values of the Mellin integral transform.

.Mf /.p/ D F.p/ D C1R n/n f.t/ f .p/ D f.t/tp1dt Conditions 0 t .p/ 1. e p Re >0;Re p>0 t2 .p=2/ 2. e 2 p=2 Re >0;Re p>0 .p/ . p/ 3. .t C 1/ . / 0

.p/ .ap/ .bp/ .c/ a; b; c 2 C;c 62 Z0 8. 2F1.a; bI cIt/ .cp/ .a/ .b/ 0

C.5 Historical and Bibliographical Notes

Here we present a few historical remarks on the early development of the integral transform method. For an extended historical exposition we refer to [DebBha07, Ch. 1]. The Fourier integral theorem was originally stated in J. Fourier’s famous treatise entitled La Théorie Analytique da la Chaleur (1822), and its deep significance was recognized by mathematicians and mathematical physicists. This theorem is one of the most monumental results of modern mathematical analysis and has widespread physical and engineering applications. Fourier’s treatise provided the modern mathematical theory of heat conduction, Fourier series, and Fourier integrals with applications. He gave a series of examples before stating that an arbitrary function defined on a finite interval can be expanded in terms of a trigonometric series which is now universally known as the Fourier series. In an attempt to extend his new ideas 314 C Integral Transforms to functions defined on an infinite interval, Fourier discovered an integral transform and its inversion formula which are now well-known as the Fourier transform and the inverse Fourier transform. However, this celebrated idea of Fourier was known to P.S. Laplace and A.L. Cauchy as some of their earlier work involved this transformation. On the other hand, S.D. Poisson also independently used the method of transform in his research on the propagation of water waves. It was the work of Cauchy that contained the exponential form of the Fourier Integral Theorem. Cauchy also introduced the functions of the operator D:

ZC1 C1Z 1 .D/f.x/ D .i/ei.xy/f.y/dyd; 2 1 1 which led to the modern form of operator theory. The birth of the operational method in its popularization as a powerful method for solving differential equations is probably due to O. Heaviside (see, e.g., [Hea93]). He developed this technique as a purely algebraic one, using and developing the classical ideas of Fourier, Laplace and Cauchy. An extended use of the machinery of complex analytic functions in this theory was proposed by T.J. Bromwich (see, e.g., [Bro09, Bro26]). In this direction several types of integral transform appeared. In particular, elaborating the ideas of B. Riemann, Mellin introduced a new type of integral transform which was later given his name. It turned out that (see [Sla66, Mari83]) the Mellin transform is highly suitable for the study of properties of a wide class of special functions, namely G-andH-functions [MaSaHa10]. In [Mari83], a calculation method for integrals with a ratio of products of Gamma functions was developed. This method is based on the properties of the Mellin transform. Many results were obtained by different authors due to a combination of the complex analytic approach and results from the theory of special functions which rapidly developed in the ﬁrst part of the twentieth century. These results led to the theory of operational calculus in its modern form. We mention here several treaties on integral transform theory [Dav02,DebBha07,DitPru65,Doe74,Mik59,Sne74,Tit86,Wid46]. Applications of the integral transform method are presented in many books on integral and differential equations (see, e.g., [AnKoVa93, Boa83, PolMan08]), in particular those related to the fractional calculus [Die10, KiSrTr06, SaKiMa93]. Tables of integral transforms (see [BatErd54a,GraRyz00,Mari83,Obe74]) constitute a very useful source for applications.

C.6 Exercises

C.6.1. Evaluate the Fourier transform of the functions ([DebBha07, p. 119]) n o n o at2 2 t 2 .a/f .t/ D texp 2 ; a > 0 .b/f .t/ D t exp 2 : C.6 Exercises 315

C.6.2. Solve the following integral equations with respect to the unknown function y.t/ ([DebBha07, p. 121])

ZC1 .a/ .x t/y.t/dt D g.x/; 1 ZC1 .b/ exp at2 y.x t/dt D exp ax2 ;a>b>0: 1

Hint. Use the Fourier transform. C.6.3. Use the Fourier transform to solve the boundary value problem ([DebBha07, p. 128]) 2 uxx C uyy D xexp x ; 1

in the class of continuously differentiable functions such that

lim u.x; y/ D 0; 8x; 1

Answer.

ZC1 Â Ã p sin tx t 2 u.x; y/ D 2 Œ1 exp .ty/ exp dt: t 4 0

C.6.4. Find the Laplace transform of the functions ([DebBha07, p. 173]):

.a/2t C a sin at;a>0; .b/.1 2t/exp f2tg ; .c/H.t a/exp ft ag ;a>0;.d/.t a/kH.t a/;a>0;k2 N:

C.6.5. Evaluate the inverse Laplace transform of the functions ([DebBha07, p. 174])

1 1 .a/ .sa/.sb/2 ;a;b>0I .b/ s2.sa/2 ;a>0I 1 s .c/ s2.s2Ca2/ ;a>0I .d/ .s2Ca2/.s2Cb2/ ;a;b>0: 316 C Integral Transforms

C.6.6. Using the change of variables, s D c C i!, show that the inverse Laplace transformation is a Fourier transformation, that is ([DebBha07, p. 179]),

Á ZC1 ect f.t/ D L1f.s/Q .t/ D f.cQ C i!/ei!td!: 2 1

C.6.7. Calculate the Mellin transform of the functions ([DebBha07, p. 365])

.a/f.t/D H.a t/; a > 0I .b/f.t/ D t ˛eˇt ;˛;ˇ>0I

1 2 .c/f.t/D 1Ct 2 I .d/f.t/ D J0 .t/:

C.6.8. Prove that ([DebBha07, p. 365]) Â Ã 1 .p/ .n p/ M .p/ D ;a>0I .1 C at/n ap .n/ Á p 1 a np 1 M n 2 . t Jn.at// .p/ D p ;a>0;n> : 2 2 n 2 C 1 2

C.6.9. Prove the following relations of the Mellin transform to the Laplace and the Fourier transforms ([DebBha07, p. 370]): .Mf.t//.p/D Lf.et / .p/I .Mf.t//.aC i!/ D Ff.et /eat .!/:

C.6.10. Calculate the Laplace transform of the Bessel function of the ﬁrst kind J.t/ for Re >1, where ([KiSrTr06, p. 36, (1.7.34)])

X1 .1/k.z=2/2kC J .z/ D .z 2 C n .1;0I 2 C/: kŠ . C k C 1/ kD0

Answer.

L h 1 i . J.t// .s/ D p p .Re s>0/: s2 C 1 s C s2 C 1

C.6.11. Calculate the Laplace transform of the Bessel function of the second kind Y.t/ for jRe j <1;6D 0, where ([KiSrTr06, p. 36, (1.7.35)])

cos ./J .z/ J .z/ Y .z/ D . 2 C n Z/: sin C.6 Exercises 317

Answer. h i p 2 cot ./ csc ./ s C s2 C 1 L h i . Y.t// .s/ D p p .Re s>0/: s2 C 1 s C s2 C 1

C.6.12. Prove the following relation (˛; ˇ; 2 C; Re ˛>0)([KiSrTr06, p. 50, (1.10.10)]) Â Ä Â Ã Ã ˇ ˇ n ˛ˇ ˇ ˇ ˛nCˇ1 @ ˛ nŠs ˇ ˇ L t E˛;ˇ. t / .s/ D .ˇ ˇ <1/: @ .s˛ /nC1 s˛

z ˛1 ˛ C C C.6.13. Let e˛ D z E˛;˛. z /.z 2 I Re ˛>0I 2 / be the so-called ˛- Exponential function. Prove the following relation for it ([KiSrTr06, p. 52, (1.10.27)]): Â ÄÂ Ã Ã ˇ ˇ n ˇ ˇ @ t nŠ ˇ ˇ L e .s/ D .n 2 NI Re s>0I ˇ ˇ <1/: @ ˛ .s˛ /nC1 s˛

C.6.14. Calculate the Laplace transform of the Meyer G-function ([MaSaHa10, p. 52, (2.29)]) Ä ˇ ˇ 1 m;n ˇ a1;:::;ap t Gp;q at b1;:::;bq

2 C; >0;Re C min Re bj >0; 1Äj Äm c p C q jarg aj < ;c D m C n >0I Re s>0: 2 2 C.6.15. Calculate for all s 2 C,Res>0, the Laplace transform of the Fox H- function ([MaSaHa10, p. 50, (2.19)]) Ä ˇ ˇ 1 m;n ˇ .a1;˛1/;:::;.ap;˛p / t Hp;q at .b1;ˇ1/;:::;.bq;ˇq / ; a 2 C; >0; ˛ ˛>0;jarg aj < or ˛ D 0 and Re ı<1 2 Re b Re C min j >0; for ˛>0or ˛ D 0; 0; 1Äj Äm ˇj 318 C Integral Transforms

and Ä Re b Re ı C 1=2 Re C min j C >0; for ˛ D 0; < 0; 1Äj Äm ˇj

where

Xn Xp Xm Xq ˛ D ˛j ˛j C ˇj ˇj I j D1 j DnC1 j D1 j DmC1 q p q p X X p q X X ı D b a C I D ˇ ˛ : j j 2 j j j D1 j D1 j D1 j D1

C.6.16. Calculate the Laplace and Mellin transforms of the classical Wright function .˛;ˇI t/, where ([KiSrTr06, p. 55, (1.11.6–7)])

X1 zk .˛;ˇI z/ D : kŠ .˛k C ˇ/ kD0

Answers. Â Ã 1 1 .L .˛;ˇI t//.s/ D E .˛ > 1I ˇ 2 CI Re s>0/I s ˛;ˇ s .s/ .M .˛;ˇI t//.s/ D .˛ > 1I ˇ 2 CI Re s>0/: .ˇ ˛s/

C.6.17. Find the Mellin transform of the Gauss hypergeometric function 2F1.a; bI cIt/ (a; b; c 2 C)([MaSaHa10, p. 46, Ex. 2.1]). Answer.

.s/ .a s/ .b s/ .c/ .M F .a; bI cIt//.s/ D 2 1 .c s/ .a/ .b/ .min fRe a; Re bg > Re s>0/:

C.6.18. Calculate the Mellin transform of the Meyer G-function ([MaSaHa10, p. 48, (2.9)]) Ä ˇ ˇ m;n ˇ a1;:::;ap Gp;q at b1;:::;bq c p C q a 2 C; jarg aj ;c D m C n >0I 2 2

s 2 C; min Re bj < Re s<1 max Re aj : 1Äj Äm 1Äj Än Appendix D The Mellin–Barnes Integral

D.1 Deﬁnition: Contour of Integration

In the modern theory of special functions it has become customary to call to an integral of the type (see, e.g., [ParKam01]) Z 1 I.z/ D f.s/zs ds; (D.1.1) 2i L a Mellin–Barnes integral. Here the density function f.s/ is usually a solution to a certain ordinary differential equation with polynomial coefficients. Thus this integral is similar to the Mellin transform applied to special types of originals (see, e.g., [Mari83]). The most crucial part of this definition is the choice of the contour of integration. The contour L is usually either a loop in the complex s plane, a vertical line indented to avoid certain poles of the integrand, or a curve midway between these two, in the sense of avoiding certain poles of the integrand and tending to infinity in certain fixed directions (see, e.g., [ParKam01]). A short introduction to the theory of Mellin–Barnes integrals is given in [Bat-1, pp. 49–50]. In order to be more precise we consider a density function f.s/of the type

A.s/B.s/ f.s/ D ; (D.1.2) C.s/D.s/ where A; B; C; D are products of Gamma functions depending on parameters. Such integrals appear, in particular, in the representation of the solution to the hypergeometric equation

d2u du z.1 z/ C Œc .b C a 1/z abu D 0; (D.1.3) dz2 dz

© Springer-Verlag Berlin Heidelberg 2014 319 R. Gorenﬂo et al., Mittag-Lefﬂer Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2 320 D The Mellin–Barnes Integral i.e. in the representation of the Gauss hypergeometric function

X1 .a/ .b/ F.a;bI cI z/ D n n zn; .c/ nD0 n and in the representation of the solution to a more general equation, namely, the generalized hypergeometric equation 2 3 p Â Ã q Â Ã Y d d Y d 4z z C a z z C b 1 5 u.z/ D 0; (D.1.4) dz j dz dz k j D0 kD0 i.e. in the representation of the generalized hypergeometric function

X1 .a / :::.a / F .a ;:::;a I b ;:::;b I z/ D 1 n p n zn: p q 1 p 1 q .b / :::.b / nD0 1 n q n

In [Mei36] more general classes of transcendental functions were introduced via a generalization of the Gauss hypergeometric function presented in the form of a series (commonly known now as Meijer G-functions). Later this deﬁnition was replaced by the Mellin–Barnes representation of the G-function Ä ˇ Z ˇ 1 m;n ˇ a1;:::;ap Gm;n s Gp;q z ˇ D p;q .s/z ds; (D.1.5) b1;:::;bq 2i L where L is a suitably chosen path, z 6D 0, zs WD expŒs.ln jzjCiarg z/ with a single valued branch of arg z, and the integrand is deﬁned as Q Q m .b s/ n .1 a C s/ Gm;n Q kD1 k j DQ1 j p;q .s/ D q p : (D.1.6) kDmC1 .1 bk C s/ j DnC1 .aj s/

In (D.1.6) the empty product is assumed to be equal to 1 (the empty product convention), the parameters m; n; p; q satisfy the relation 0 Ä n Ä q, 0 Ä n Ä p, and the complex numbers aj , bk are such that no pole of .bk s/;k D 1;:::;m, coincides with a pole of .1 aj C s/;j D 1;:::;n. Let

q p 1 X X ı D m C n .p C q/; D b a : 2 k j kD1 j D1

The contour of integration L in (D.1.5) can be of the following three types (see [Mari83], [Kir94, p. 313]): D.1 Deﬁnition: Contour of Integration 321

(i) L D Li1, which starts at i1 and terminates at Ci1, leaving to the right all poles of -functions .bk s/;k D 1;:::;m, and leaving to the left all poles of -functions .1 aj C s/;j D 1;:::;n.Integral(D.1.5)converges for ı>0, jarg zj <ı.Ifjarg zjDı, ı 0, then the integral converges absolutely when p D q,Re<1,andwhenp 6D q if .q p/Res> 1 Re C 1 2 .q p/ as Ims !˙1. (ii) L D LC1, which starts at '1 C i1,terminatesat'2 C i1, 1 <'1 < '2 < C1, encircles once in the negative direction all poles of the -functions .bk s/;k D 1;:::;m, but no pole of the -functions .1aj Cs/;j D 1; :::;n. Integral (D5) converges if q 1 and either pqor p D q and jzj >1.

Since the generalized hypergeometric function pFq .a1;:::;apI b1;:::;bqI z/ can be considered as a special case of the Meijer G-functions, the function pFq.a1;:::;apI b1;:::;bq I z/ also possesses a representation via a Mellin–Barnes integral

Qq Ä ˇ .bk/ ˇ kD1 1;p ˇ .1 a1/;:::;.1 ap/ pFq.a1;:::;apI b1;:::;bqI z/ D Qp Gp;qC1 z ˇ ; 0; .1 b1/;:::;.1 bq/ .aj / j D1 (D.1.7) where Ä ˇ ˇ 1;p ˇ .1 a1/;:::;.1 ap/ Gp;qC1 z ˇ 0; .1 b1/;:::;.1 bq/

CZi1 1 .a1 C s/::: .ap C s/ .s/ D .z/s ds; (D.1.8) 2i .b1 C s/::: .bq C s/ i1

aj 6D 0; 1;:::I j D 1;:::;pIjarg .1 iz/j <:

Although the Meijer G-functions are quite general in nature, there still exist examples of special functions, such as the Mittag-Lefﬂer and the Wright functions, which do not form particular cases of them. A more general class which includes those functions can be obtained by introducing the Fox H-functions [Fox61], whose representation in terms of the Mellin–Barnes integral is a straightforward generalization of that for the G-functions. To introduce it one needs to add to the sets of the complex parameters aj and bk the new sets of positive numbers ˛j and ˇk 322 D The Mellin–Barnes Integral with j D 1;:::;p, k D 1;:::;q, and to replace in the integral of (D.1.5)thekernel p;q Gm;n.s/ by the new one Q Q m .b ˇ s/ n .1 a C ˛ s/ Hm;n Q kD1 k k j DQ1 j j p;q .s/ D q p : (D.1.9) kDmC1 .1 bk C ˇks/ j DnC1 .aj ˛j s/

The Fox H-functions are then deﬁned in the form " ˇ # ˇ Z ˇ .a ;˛ /p 1 p;q p;q ˇ j j j D1 Hm;n s Hm;n.z/ D Hm;n.z/ z ˇ q D p;q .s/z ds: (D.1.10) .bk;ˇk/ 2i kD1 L

A representation of the type (D.1.10) is usually called a Mellin–Barnes integral representation. The convergence questions for these integrals are completely discussed in [ParKam01, Sect. 2.4]. For further information we refer the reader to the treatises on Fox H-functions by Mathai, Saxena and Haubold [MaSaHa10], Srivastava, Gupta and Goyal [SrGuGo82], Kilbas and Saigo [KilSai04]andthe references therein.

D.2 Asymptotic Methods for the Mellin–Barnes Integral

The asymptotics of the Mellin–Barnes integral is based on the following two general lemmas on the expansion of quotients of Gamma functions as inverse factorial expansions (see [Wri40b, Wri40c, Bra62]). Let us consider the quotient of products of Gamma functions

Qp .˛j s C aj / j D1 P.s/ D Qq : (D.2.1) .ˇk s C bk/ kD1

Deﬁne the parameters 8 ˆ Qp Qq ˆ ˛j ˇk ˆ h D ˛j ˇk ; ˆ j D1 kD1 < Pp Pq 1 0 # D aj bk C 2 .q p C 1/; # D 1 #; (D.2.2) ˆ j D1 ˆ kD1 ˆ Pq Pp :ˆ D ˇk ˛j : kD1 j D1

The following lemma presents the inverse factorial expansions for the functions P.s/ (see [Wri40c, Bra62]). D.2 Asymptotic Methods for the Mellin–Barnes Integral 323

Lemma D.1 ([ParKam01,p.39]). Let M be a positive integer and suppose >0. Then there exist numbers Ar (0 Ä r Ä M 1), independent of s and M , such that the function P.s/ in (D.2.1) possesses the inverse factorial expansion given by ( ) MX1 A .s/ P.s/ D .h/s r C M ; (D.2.3) .sC #0 C r/ .sC #0 C M/ rD1 where the parameters h; and #0 are deﬁned in (D.2.2). In particular, the coefﬁcient A0 has the value

Yp Yq 1=2.pqC1/ 1=2# aj 1=2 1=2bk A0 D .2/ ˛j ˇk : (D.2.4) j D1 kD1

The remainder function M .s/ is analytic in s except at the points s D.aj Ct/= ˛j , t D 0; 1; 2; : : : (1 Ä j Ä p), where P.s/ has poles, and is such that

M .s/ D O.1/ for jsj!1uniformly in jargsjÄ ", ">0. Let now Q.s/ be another type of quotient of products of Gamma functions:

Qq .1 bk C ˇks/ kD1 Q.s/ D Qp : (D.2.5) .1 aj C ˛j s/ j D1

Let the parameters h; #; #0; be deﬁned in the same manner as in (D.2.2). The following lemma presents the corresponding inverse factorial expansions for the functions Q.s/ (see [Wri40c, Bra62]). Lemma D.2 ([ParKam01,p.39]). Let M be a positive integer and suppose >0. Then there exist numbers Ar (0 Ä r Ä M 1), independent of s and M , such that the function Q.s/ in (D.2.5) possesses the inverse factorial expansion given by ( ) .h/s MX1 Q.s/ D .1/r A .sC # r/C .s/ .s C # M/ ; .2/pqC1 r M rD1 (D.2.6) where the parameters h; and #0 are deﬁned in (D.2.2). The remainder function M .s/ is analytic in s except at the points s D .bk 1 t/=ˇk, t D 0; 1; 2; : : : (1 Ä k Ä q), where Q.s/ has poles, and is such that

M .s/ D O.1/ for jsj!1uniformly in jargsjÄ ", ">0. 324 D The Mellin–Barnes Integral

An algebraic method for determining the coefﬁcients Ar in (D.2.3)and(D.2.6) is presented in [ParKam01, pp. 46–49].

D.3 Historical and Bibliographical Notes

As a historical note, we point out that “Mellin–Barnes integrals” are named after the two authors (namely Hj. Mellin and E.W. Barnes) who in the early 1910s developed the theory of these integrals, using them for a complete integration of the hypergeometric differential equation. However, these integrals were first used in 1888 by Pincherle, see, e.g., [MaiPag03]. Recent treatises on Mellin–Barnes integrals are [Mari83]and[ParKam01]. In the classical treatise on Bessel functions by Watson [Wat66, p. 190], we read “By using integrals of a type introduced by Pincherle and Mellin, Barnes has obtainedrepresentationsofBesselfunctions...”. Salvatore Pincherle (1853–1936) was Professor of Mathematics at the University of Bologna from 1880 to 1928. He retired from the University just after the International Congress of Mathematicians which he had organized in Bologna, following the invitation received at the previous Congress held in Toronto in 1924. He wrote several treatises and lecture notes on Algebra, Geometry, and Real and Complex Analysis. His main book related to his scientific activity is entitled “Le Operazioni Distributive e loro Applicazioni all’Analisi”; it was written in collaboration with his assistant, Dr. Ugo Amaldi, and was published in 1901 by Zanichelli, Bologna. Pincherle can be considered one of the most prominent founders of Functional Analysis, and was described as such by J. Hadamard in his review lecture “Le développement et le rôle scientifique du Calcul fonctionnel”, given at the Congress of Bologna (1928). A description of Pincherle’s scientific works requested from him by Mittag-Leffler, who was the Editor of the prestigious journal Acta Mathematica, appeared (in French) in [Pin25]. A collection of selected papers (38 from 247 notes plus 24 treatises) was edited by Unione Matematica Italiana (UMI) on the occasion of the centenary of his birth, and published by Cremonese, Roma 1954. S. Pincherle was the first President of UMI, from 1922 to 1936. Here we point out that S. Pincherle’s 1888 paper (in Italian) on the Generalized Hypergeometric Functions led him to introduce what was later called the Mellin– Barnes integral to represent the solution of a generalized hypergeometric differential equation investigated by Goursat in 1883. Pincherle’s priority was explicitly recognized by Mellin and Barnes themselves, as reported below. In 1907 Barnes, see p. 63 in [Barn07b], wrote: “The idea of employing contour integrals involving gamma functions of the variable in the subject of integration appears to be due to Pincherle, whose suggestive paper was the starting point of the investigations of Mellin (1985) [Mel95] though the type of contour and its use can be traced back to Riemann.” In 1910 Mellin, see p. 326ff in [Mel10], devoted a section (§10: Proof of Theorems of Pincherle) to revisit the original work of Pincherle; in D.4 Exercises 325 particular, he wrote “Before we prove this theorem, which is a special case of a more general theorem of Mr. Pincherle, we want to describe more closely the lines L over which the integration is preferably to be carried out.” [free translation from German]. The Mellin–Barnes integrals are the essential tools for treating the two classes of higher transcendental functions known as G-and H-functions, introduced by Meijer [Mei46]andFox[Fox61] respectively, so Pincherle can be considered their precursor. For an exhaustive treatment of the Mellin–Barnes integrals we refer to the recent monograph by Paris and Kaminski [ParKam01].

D.4 Exercises

D.4.1. The exponential function can be represented in terms of the Mellin–Barnes integral ([ParKam01, p. 67])

cZCi1 1 ez D .s/zs ds.c>0/: 2i ci1

Find the domain of convergence of the integral in the right-hand side of this representation. D.4.2. Prove the following Mellin–Barnes integral representation of the hypergeometric function ([ParKam01, p. 68]) Z .a/ .b/ 1 .sC a/ .s C b/ F .a; bI cI z/ D .s/ .z/s ds; .c/ 2 1 2i .sC c/ Li1

where the contour of integration Li1 is a vertical line, which starts at i1 ends at Ci1 and separates the poles of the Gamma function .s/ from those of the Gamma functions .sC a/, .s C b/ for a and b 6D 0; 1; 2;:::. Find the domain of convergence of the integral in the above representation. If the vertical line Li1 is replaced by the contour L1, then show that the domain of convergence coincides with an open unit disk. D.4.3. Find the domain of convergence of the following Mellin–Barnes integral ([ParKam01,p.68]): Z 1 .a s=2/ . s C b/ .c C s=4/ .z/s ds; 2i .d C s=4/ .s C 1/ LC1 326 D The Mellin–Barnes Integral

where the parameters a; b; c and and the contour of integration LC1 are such that the poles of .as=2/ are separated from those of . sCb/ and .cC s=4/. D.4.4. Prove that the following Mellin–Barnes integral representation ([ParKam01, p. 109])

cZCi1 .a/ 1 .s/ .s C a/ F .aI bI cI z/ D .z/sds .b/1 1 2i .sC b/ ci1

is valid for all ﬁnite values of c provided that the contour of integration can be deformed to separate the poles of .s/ and .sC a/ (which is always possible when a is not a negative integer or zero). Here 1F1.aI bI cI z/ is the conﬂuent hypergeometric function

X1 .a/ zn F .aI bI cI z/ D n ;.jzj < 1/: 1 1 .b/ nŠ nD n

D.4.5. For the incomplete gamma function ([ParKam01, p. 113])

Zz .a;z/ D t a1et dt.Re a>0/

prove the representation

cZCi1 1 .s/ .a;z/ D .a/C zsCads 2i s C a ci1

where c

Z1 .a;z/ D t a1et dt z

prove the representation

cZCi1 1 .sC a/ .a;z/ D zsds 2i s ci1

for all z; jarg zj=2. Appendix E Elements of Fractional Calculus

E.1 Introduction to the Riemann–Liouville Fractional Calculus

As is customary, let us take as our starting point for the development of the so-called Riemann–Liouville fractional calculus the repeated integral Z Z Z x xn1 x1 n N IaC .x/ WD ::: .x0/ dx0 ::: dxn1 ;aÄ x1 and b ÄC1. The function .x/ is assumed to be well-behaved; for this it sufﬁces that .x/ is locally integrable in the interval Œa; b/, meaning in particular that a possible singular behaviour at x D a does not destroy integrability. It is well known that the above formula provides an n-fold primitive n.x/ of .x/, precisely that primitive which vanishes at x D a jointly with its derivatives of order 1;2;:::n 1: We can re-write this n-fold repeated integral by a convolution-type formula (often attributed to Cauchy) as, Z x n 1 n1 IaC .x/ D .x / ./d; aÄ x

In a natural way we are now led to extend the formula (E.1.2) from positive integer values of the index n to arbitrary positive values ˛;thereby using the relation .n 1/Š D .n/:So, using the Gamma function, we deﬁne the fractional integral of order ˛ as Z x ˛ 1 ˛1 IaC .x/ WD .x / ./d; a0: (E.1.3) .˛/ a

© Springer-Verlag Berlin Heidelberg 2014 327 R. Gorenﬂo et al., Mittag-Lefﬂer Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2 328 E Elements of Fractional Calculus

n N We remark that the values IaC .x/ with n 2 are always ﬁnite for a Ä x0are ﬁnite for a0we obtain the dual form of the fractional integral (E.1.3), i.e. Z b ˛ 1 ˛1 Ib .x/ WD . x/ ./d; a0: (E.1.5) .˛/ x

˛ Now it may happen that the limit (if it exists) of Ib .x/ for x ! b ,thatwe ˛ denote by Ib .b /, is inﬁnite. ˛ ˛ We refer to the fractional integrals IaC and Ib as progressive or right-handed and regressive or left-handed, respectively. Let us point out the fundamental property of the fractional integrals, namely the additive index law (semigroup property,) according to which

˛ ˇ ˛Cˇ ˛ ˇ ˛Cˇ IaC IaC D IaC ;Ib Ib D Ib ;˛;ˇ 0; (E.1.6)

0 0 I where, for complementation, we have deﬁned IaC D Ib WD (Identity operator) 0 0 which means IaC .x/ D Ib .x/ D .x/. The proof of (E.1.6) is based on Dirichlet’s formula concerning the change of the order of integration and the use of the Beta function in terms of the Gamma function. We note that the fractional integrals (E.1.3)and(E.1.5) contain a weakly singular kernel only when the order is less than one. We can now introduce the concept of fractional derivative based on the fundamental property of the common derivative of integer order n

dn Dn .x/ D .x/ D .n/.x/ ; a < x < b dxn E.1 Introduction to the Riemann–Liouville Fractional Calculus 329 to be the left inverse operator of the progressive repeated integral of the same n order n, IaC .x/. In fact, it is straightforward to recognize that the derivative of any integer order n D 0; 1; 2; : : : satisﬁes the following composition rules with n n respect to the repeated integrals of the same order n, IaC .x/ and Ib .x/ W ( n n D IaC .x/ D .x/; P C a

In view of the above properties we can define the progressive/regressive fractional derivatives of order ˛>0provided that we first introduce the positive integer m such that m 1<˛Ä m. Then we define the progressive/regressive fractional ˛ ˛ derivative of order ˛, DaC=Db,astheleft inverse of the corresponding fractional ˛ ˛ integral IaC=Ib. As a consequence of the previous formulas (E.1.2)–(E.1.8)the fractional derivatives of order ˛ turn out to be defined as follows ( D˛ .x/ WD Dm I m˛ .x/; aC aC a

In fact, for the progressive/regressive derivatives we get ( D˛ I ˛ D Dm I m˛ I ˛ D Dm I m D I ; aC aC aC aC aC (E.1.10) ˛ ˛ m m m˛ ˛ m m m I Db Ib D .1/ D Ib Ib D .1/ D Ib D :

0 0 I For complementation we also deﬁne DaC D Db WD (Identity operator). When the order is not an integer (m 1<˛

We stress the fact that the “proper” fractional derivatives (namely when ˛ is non-integer) are non-local operators being expressed by ordinary derivatives of convolution-type integrals with a weakly singular kernel, as is evident from (E.1.11) 330 E Elements of Fractional Calculus and (E.1.12). Furthermore, they do not necessarily obey the analogue of the “semigroup” property of the fractional integrals: in this respect the initial point a 2 R (or the end point b 2 R) plays a “disturbing” role. We also note that when the order ˛ tends to both the end points of the interval .m 1; m/ we recover the common derivatives of integer order (unless of the sign for the regressive derivative).

E.2 The Liouville–Weyl Fractional Calculus

We can also deﬁne the fractional integrals over unbounded intervals and, as left inverses, the corresponding fractional derivatives. If the function .x/ is locally integrable in 1 0: (E.2.1) .˛/ 1

Analogously, if the function .x/ is locally integrable in 1 Ä a0: (E.2.2) .˛/ x

Note the kernel . x/˛1 for (E.2.2). The names of Liouville and Weyl are here adopted for the fractional integrals (E.2.1)and(E.2.2), respectively, following a standard terminology, see e.g. Butzer and Westphal [ButWes00],basedonhistorical reasons.1 We note that a sufﬁcient condition for the integrals in (E.2.1)and(E.2.2)to converge is that

.x/D O.jxj˛ /;>0;x! 1; respectively. Integrable functions satisfying these properties are sometimes referred to as functions of Liouville class (for x !1), and of Weyl class (for x !C1), respectively, see Miller and Ross [MilRos93]. For example, power functions jxjı

1In fact, Liouville considered in a series of papers from 1832 to 1837 the integrals of progressive type (E.2.1), see, e.g., [Lio32a, Lio32a, Lio37]. On the other hand, H. Weyl [Wey17] arrived at the regressive integrals of type (E.2.2) indirectly by deﬁning fractional integrals suitable for periodic functions. E.2 The Liouville–Weyl Fractional Calculus 331 with ı>˛>0and exponential functions ecx with c>0are of Liouville class (for x !1). For these functions we obtain 8 ˆ ˛ ı .ı˛/ ıC˛ < IC jxj D .ı/ jxj ; ˆ ı >˛ >0; x <0; (E.2.3) : ˛ ı .ıC˛/ ı˛ DC jxj D .ı/ jxj ; and 8 ˛ cx ˛ cx < IC e D c e ; R : c>0;x2 : (E.2.4) ˛ cx ˛ cx DC e D c e ;

For the Liouville and Weyl fractional integrals we can also state the corresponding semigroup property

˛ ˇ ˛Cˇ ˛ ˇ ˛Cˇ IC IC D IC ;I I D I ;˛;ˇ 0; (E.2.5)

0 0 I where, for complementation, we have defined IC D I WD (Identity operator). For more details on Liouville–Weyl fractional integrals we refer to Miller [Mil75], Samko, KIlbas and Marichev [SaKiMa93] and Miller and Ross [MilRos93]. For the definition of the Liouville–Weyl fractional derivatives of order ˛ we follow the scheme adopted in the previous section for bounded intervals. Having introduced the positive integer m so that m 1<˛Ä m we define 8 ˛ m m˛ < DC .x/ WD D IC .x/; 1

0 0 I with DC D D D : In fact we easily recognize using (E.2.5)and(E.2.6)the fundamental property

˛ ˛ I m ˛ ˛ DC IC D D .1/ D I : (E.2.7)

The explicit expressions for the “proper” Liouville and Weyl fractional derivatives (m 1<˛

Because of the unbounded intervals of integration, fractional integrals and derivatives of Liouville and Weyl type can be (successfully) handled via the Fourier transform and the related theory of pseudo-differential operators, that, as we shall see, simplifies their treatment. For this purpose, let us now recall our notations and the relevant results concerning the Fourier transform. Let Z C1 ./O D F f .x/I g D e Cix .x/dx; 2 R ; (E.2.10) 1 be the Fourier transform of a sufficiently well-behaved function .x/,andlet n o Z 1 C1 .x/ D F 1 ./O I x D e ix ./O d; x2 R ; (E.2.11) 2 1 denote the inverse Fourier transform.2 In this framework we also consider the class of pseudo-differential operators of which the ordinary repeated integrals and derivatives are special cases. A pseudo- differential operator A, acting with respect to the variable x 2 R, is defined through its Fourier representation, namely Z C1 e ix A .x/dx D A./O ./;O (E.2.12) 1 where A./O is referred to as the symbol of A. An often applicable practical rule is A./O D A e ix e Cix ;2 R : (E.2.13)

If B is another pseudo-differential operator, then we have

AB./b D A./O B./:O (E.2.14)

F For the sake of convenience we adopt the notation $ to denote the juxtaposition of a function with its Fourier transform and that of a pseudo-differential operator with its symbol, i.e.

F F .x/$ ./;O A$ A:O (E.2.15)

2In the ordinary theory of the Fourier transform the integral in (E.2.10) is assumed to be a “Lebesgue integral” whereas the one in (E.2.11) can be the “principal value” of a “generalized integral”. In fact, .x/ 2 L1.R/, necessary for writing (E.2.10), is not sufﬁcient to ensure ./O 2 L1.R/. However, we allow for an extended use of the Fourier transform which includes Dirac-type generalized functions: then the above integrals must be properly interpreted in the framework of the theory of distributions. E.2 The Liouville–Weyl Fractional Calculus 333

We now consider the pseudo-differential operators represented by the Liouville– Weyl fractional integrals and derivatives. Of course we assume that the integrals in their deﬁnitions are in a proper sense, in order to ensure that the resulting functions of x can be Fourier transformable in the ordinary or generalized sense. The symbols of the fractional Liouville–Weyl integrals and derivatives can easily be derived according to ( Ic˛ D . i/˛ Djj˛ e ˙i.sgn/˛=2 ; ˙ (E.2.16) b˛ C˛ C˛ i.sgn/˛=2 D˙ D . i/ Djj e :

Based on an idea of Marchaud, see e.g. Marchaud [Marc27], Samko, Kilbas and Marichev [SaKiMa93], Hilfer [Hil97], we now give purely integral expressions for ˛ D˙ which are alternative to the integro-differential expressions (E.2.8)and(E.2.9). We limit ourselves to the case 0<˛<1:Let us ﬁrst consider from Eq. (E.2.8) the progressive derivative

d D˛ D I 1˛ ; 0<˛<1: (E.2.17) C dx C We have, see Hilfer [Hil97],

d D˛ .x/ D I 1˛ .x/ C dx C Z 1 d x D .x /˛ ./d .1 ˛/ dx 1 Z 1 d 1 D ˛ .x /d .1 ˛/ dx 0 Z Ä Z ˛ 1 1 d D 0.x / ; 1C˛ d .1 ˛/ 0 so that, interchanging the order of integration Z ˛ 1 .x/ .x / D˛ .x/ D ; 0<˛<1: C 1C˛ d (E.2.18) .1 ˛/ 0

Here 0 denotes the ﬁrst derivative of with respect to its argument. The coefﬁcient in front of the integral in (E.2.18) can be re-written, using known formulas for the Gamma function, as

˛ 1 sin ˛ D D .1C ˛/ : (E.2.19) .1 ˛/ .˛/ 334 E Elements of Fractional Calculus

Similarly we get for the regressive derivative

d D˛ D I 1˛ ;0<˛<1; (E.2.20) dx Z ˛ 1 .x/ .x C / D˛ .x/ D ; 0<˛<1: 1C˛ d (E.2.21) .1 ˛/ 0

Similar results can be given for ˛ 2 .m 1; m/ ; m 2 N :

E.3 The Abel–Riemann Fractional Calculus

RC In this section we consider sufﬁciently well-behaved functions .t/ (t 2 0 ) with Laplace transform deﬁned as Z 1 st Q .s/ D L f .t/I sg D e .t/ dt; Re.s/ > a ; (E.3.1) 0 where a denotes the abscissa of convergence. The inverse Laplace transform is then given as Z ˚ « 1 .t/ D L1 Q .s/I t D e st Q .s/ ds; t >0; (E.3.2) 2i Br

3 where Br is a Bromwich path, namely f i1; C i1g with >a . It may be convenient to consider .t/ as a “causal” function in R, i.e. vanishing L for all t<0:For the sake of convenience we adopt the notation $ to denote the juxtaposition of a function with its Laplace transform, with its symbol, i.e.

L .t/ $ Q .s/ : (E.3.3)

E.3.1 The Abel–Riemann Fractional Integrals and Derivatives

We ﬁrst deﬁne the Abel–Riemann (A-R) fractional integral and derivative of any order ˛>0for a generic (well-behaved) function .t/ with t 2 RC :

3 C In the ordinary theory of the Laplace transform the condition .t/ 2 Lloc.R / is necessarily required, and the Bromwich integral is intended in the “principal value” sense. However, as mentioned in the previous footnote, by interpreting these integrals in the framework of the theory of distributions we include Dirac-type generalized functions. E.3 The Abel–Riemann Fractional Calculus 335

For the A-R fractional integral (of order ˛)wehave Z 1 t J ˛ .t/ WD .t /˛1 . / d ; t>0 ˛>0: (E.3.4) .˛/ 0

For complementation we put J 0 WD I (Identity operator), as can be justiﬁed by passing to the limit ˛ ! 0: The A-R integrals possess the semigroup property

J ˛ J ˇ D J ˛Cˇ ; for all ˛; ˇ 0: (E.3.5)

The A-R fractional derivative (of order ˛>0)isdeﬁnedastheleft-inverse operator of the corresponding A-R fractional integral (of order ˛>0), i.e.

D˛ J ˛ D I : (E.3.6)

Therefore, introducing the positive integer m such that m 1<˛Ä m and noting that .Dm J m˛/J˛ D Dm .J m˛ J ˛/ D Dm J m D I ; we deﬁne

D˛ WD Dm J m˛ ;m 1<˛Ä m; (E.3.7) i.e. ( R 1 dm t . / m ˛C1m d ; m 1<˛

For complementation we put D0 WD I : For ˛ ! m we thus recover the standard derivative of order m but the integral formula loses its meaning for ˛ D m: By using the properties of the Eulerian Beta and Gamma functions it is easy to show the effect of our operators J ˛ and D˛ on the power functions: we have 8 ˆ J ˛t D .C1/ t C˛ ; <ˆ .C1C˛/ D˛ t D .C1/ t ˛ ; (E.3.9) ˆ .C1˛/ :ˆ t>0;˛ 0; >1:

These properties are of course a natural generalization of those known when the order is a positive integer. Note the remarkable fact that the fractional derivative D˛ .t/ is not zero for the constant function .t/ Á 1 if ˛ 62 N : In fact, the second formula in (E.3.9) with D 0 teaches us that

t ˛ D˛ 1 D ;˛ 0; t >0: (E.3.10) .1 ˛/ 336 E Elements of Fractional Calculus

This, of course, is identically 0 for ˛ 2 N, due to the poles of the Gamma function at the points 0; 1; 2;:::.

E.4 The Caputo Fractional Calculus

E.4.1 The Caputo Fractional Derivative

An alternative definition of the fractional derivative was introduced in the late sixties by Caputo [Cap67, Cap69] and was soon adopted in physics to deal with long-memory visco-elastic processes by Caputo and Mainardi, see [CapMai71a, CapMai71b], and, for a more recent review, Mainardi [Mai97]. ˛ In this case the fractional derivative, denoted by D ; is defined by exchanging the operators J m˛ and Dm in the classical definition (E.3.7), i.e.

˛ m˛ m D WD J D ;m 1<˛Ä m: (E.4.1)

In the literature, following its appearance in the book by Podlubny [Pod99], this derivative is known simply as the Caputo derivative. Based on (E.4.1)wehave 8 R < 1 t .m/. / C d ; m 1<˛

For m1<˛

˛ m m˛ m˛ m ˛ D .t/ D D J .t/ ¤ J D .t/ D D .t/ ; (E.4.3) unless the function .t/ along with its ﬁrst m 1 derivatives vanishes at t D 0C.In fact, assuming that the passage of the m-th derivative under the integral is legitimate, one recognizes that, for m 1<˛0

mX1 t k˛ D˛ .t/ D D˛ .t/ C .k/.0C/: (E.4.4) .k ˛ C 1/ kD0

As noted by Samko, Kilbas and Marichev [SaKiMa93] and Butzer and Westphal [ButWes00] the identity (E.4.4) was considered by Liouville himself (but not used as an alternative deﬁnition of the fractional derivative). E.4 The Caputo Fractional Calculus 337

Recalling the fractional derivative of the power functions we can rewrite (E.4.4) in the equivalent form ! mX1 t k D˛ .t/ .k/.0C/ D D˛ .t/ : (E.4.5) kŠ kD0

The subtraction of the Taylor polynomial of degree m1 at t D 0C from .t/ yields a sort of regularization of the fractional derivative. In particular, as is easily shown, according to this deﬁnition we retrieve the property that the fractional derivative of a constant is zero,

˛ D1 Á 0; ˛ >0: (E.4.6)

As a consequence of (E.4.5) we can interpret the Caputo derivative as a sort of regularization of the R-L derivative as soon as the values k .0C/ are ﬁnite; in this sense such a fractional derivative was independently introduced in 1968 by Dzherbashyan and Nersesian [DzhNer68], as pointed out in interesting papers by Kochubei, see [Koc89, Koc90]. In this respect the regularized fractional derivative is sometimes referred to as the Caputo–Dzherbashyan derivative. We now explore the most relevant differences between the two fractional derivatives (E.3.7)and(E.4.1). We agree to call (E.4.1)theCaputo fractional derivative to distinguish it from the standard A-R fractional derivative. We observe, again by looking at the second equation in (E.3.9), that D˛t ˛k Á 0; t > 0 for ˛>0,andk D 1;2;:::;m. We thus recognize the following statements about functions which for t>0 admit the same fractional derivative of order ˛;with m 1<˛Ä m, m 2 N ;

Xm ˛ ˛ ˛j D .t/ D D .t/ ” .t/ D .t/ C cj t ; (E.4.7) j D1 Xm ˛ ˛ mj D .t/ D D .t/ ” .t/ D .t/ C cj t ; (E.4.8) j D1 where the coefﬁcients cj are arbitrary constants. For the two deﬁnitions we also note a difference with respect to the formal limit as ˛ ! .m 1/C; from (E.3.7)and(E.4.1) we obtain respectively,

D˛ .t/ ! Dm J .t/D Dm1 .t/ I (E.4.9)

˛ m m1 .m1/ C D .t/ ! JD .t/ D D .t/ .0 /: (E.4.10)

We now consider the Laplace transform of the two fractional derivatives. For the A-R fractional derivative D˛ the Laplace transform, assumed to exist, requires the knowledge of the (bounded) initial values of the fractional integral J m˛ and of its 338 E Elements of Fractional Calculus integer derivatives of order k D 1;2;:::;m 1:The corresponding rule reads, in our notation,

m1 L X D˛ .t/ $ s˛ Q .s/ Dk J .m˛/ .0C/sm1k ;m1<˛Ä m: (E.4.11) kD0

For the Caputo fractional derivative the Laplace transform technique requires the knowledge of the (bounded) initial values of the function and of its integer derivatives of order k D 1;2;:::;m 1;in analogy with the case when ˛ D m:In ˛ ˛ ˛ m˛ m m m fact, noting that J D D J J D D J D ,wehave

mX1 t k J ˛ D˛ .t/ D .t/ .k/.0C/ ; (E.4.12) kŠ kD0 so we easily prove the following rule for the Laplace transform,

m1 L X ˛ ˛ Q .k/ C ˛1k D .t/ $ s .s/ .0 /s ;m 1<˛Ä m: (E.4.13) kD0

Indeed the result (E.4.13), first stated by Caputo [Cap69], appears as the “natural” generalization of the corresponding well-known result for ˛ D m: Gorenflo and Mainardi [GorMai97] have pointed out the major utility of the Caputo fractional derivative in the treatment of differential equations of fractional order for physical applications. In fact, in physical problems, the initial conditions are usually expressed in terms of a given number of bounded values assumed by the field variable and its derivatives of integer order, despite the fact that the governing evolution equation may be a generic integro-differential equation and therefore, in particular, a fractional differential equation. Nowadays all the books on Fractional Calculus after Podlubny book [Pod99] consider Caputo derivative. This derivative becomes very popular because of its importance for the theory and applications. Several applications have also been treated by Caputo himself from the seventies up to the present, see e.g. [Cap74, Cap79, Cap85, Cap89, Cap96, Cap00, Cap01].

E.5 The Riesz–Feller Fractional Calculus

The purpose of this section is to combine the Liouville–Weyl fractional integrals and derivatives in order to obtain the pseudo-differential operators considered around the 1950s by Marcel Riesz [Rie49] and William Feller [Fel52]. In particular the Riesz– Feller fractional derivatives will be used later to generalize the standard diffusion equation by replacing the second-order space derivative. In so doing we shall E.5 The Riesz–Feller Fractional Calculus 339 generate all the (symmetric and non-symmetric) Lévy stable probability densities according to our parametrization.

E.5.1 The Riesz Fractional Integrals and Derivatives

The Liouville–Weyl fractional integrals can be combined to give rise to the Riesz fractional integral (usually called the Riesz potential)oforder˛,deﬁnedas Z ˛ ˛ C1 ˛ IC .x/ C I .x/ 1 ˛1 I0 .x/ D D jx j ./d; 2 cos.˛=2/ 2 .˛/ cos.˛=2/ 1 (E.5.1) for any positive ˛ with the exclusion of odd integers for which cos.˛=2/ vanishes. The symbol of the Riesz potential turns out to be

c˛ ˛ R I0 Djj ;2 ;˛>0;˛¤ 1; 3; 5 ::: : (E.5.2)

In fact, recalling the symbols of the Liouville–Weyl fractional integrals, see Eq. (E.2.16), we obtain Ä 1 1 .Ci/˛ C .i/˛ 2 cos.˛=2/ Ic˛ C Ic˛ D C D D : C .i/˛ .Ci/˛ jj˛ jj˛

We note that, in contrast to the Liouville fractional integral, the Riesz potential has the semigroup property only in restricted ranges, e.g.

˛ ˇ ˛Cˇ I0 I0 D I0 for 0<˛ <1; 0<ˇ <1; ˛C ˇ<1: (E.5.3)

From the Riesz potential we can deﬁne by analytic continuation the Riesz ˛ fractional derivative D0 , including also the singular case ˛ D 1;by formally setting ˛ ˛ D0 WD I0 , i.e., in terms of symbols,

c˛ ˛ D0 WD jj : (E.5.4)

We note that the minus sign has been used in order to recover for ˛ D 2 the standard second derivative. Indeed, noting that

jj˛ D.2/˛=2 ; (E.5.5) we recognize that the Riesz fractional derivative of order ˛ is the opposite of the d2 ˛=2-power of the positive deﬁnite operator dx2 Â Ã d 2 ˛=2 D˛ D : (E.5.6) 0 dx2 340 E Elements of Fractional Calculus

We also note that the two Liouville fractional derivatives are related to the ˛-power d of the first order differential operator D D dx : We note that it was Bochner [Boc49] who first introduced fractional powers of the Laplacian to generalize the diffusion equation. Restricting our attention to the range 0<˛Ä 2 the explicit expression for the Riesz fractional derivative turns out to be 8 ˛ ˛ <ˆ DC .x/CD .x/ 2 cos.˛=2/ ;if˛¤ 1; D˛ .x/ D (E.5.7) 0 :ˆ DH .x/; if˛D 1; where H denotes the Hilbert transform operator defined by Z Z 1 C1 ./ 1 C1 .x / H .x/WD d D d; (E.5.8) 1 x 1 the integral understood in the Cauchy principal value sense. Incidentally, we note that H 1 DH. By using the practical rule (E.2.13) we can derive the symbol of H, namely

HO D i sgn2 R : (E.5.9)

The expressions in (E.5.7) can easily be veriﬁed by manipulating with symbols of “good” operators as below

( ˛ ˛ .i/ C.Ci/ Djj˛ ;if˛¤ 1; c˛ b˛ ˛ 2 cos.˛=2/ D0 DI0 Djj D Ci isgn D sgn Djj ;if˛D 1:

In particular, from (E.5.7) we recognize that Â Ã 1 1 d2 d2 d2 d D2 D D2 C D2 D C D ; but D1 ¤ : 0 2 C 2 dx2 dx2 dx2 0 dx

˛ We also recognize that the symbol of D0 (0<˛Ä 2) is just the cumulative function (logarithm of the characteristic function) of a symmetric Lévy stable probability distribution function, see e.g. Feller [Fel71], Sato [Sat99]. We introduce the “illuminating” notation introduced by Zaslavsky, see e.g. Saichev and Zaslavsky [SaiZas97] to denote our Liouville and Riesz fractional derivatives d˛ d˛ D˛ D ;D˛ D ;0<˛Ä 2: (E.5.10) ˙ d.˙x/˛ 0 djxj˛ E.5 The Riesz–Feller Fractional Calculus 341

Recalling from (E.5.7) the fractional derivative in Riesz’s sense

D˛ .x/ C D˛ .x/ D˛ .x/ WD C ; 0<˛<1;1<˛<2; 0 2 cos.˛=2/ and using (E.2.18)and(E.2.21) we get for it the following regularized representation, valid also for ˛ D 1; Z sin .˛=2/ 1 .x C / 2 .x/ C .x / D˛ .x/ D .1C ˛/ ; 0 1C˛ d 0 0<˛<2: (E.5.11) We note that Eq. (E.5.11) has recently been derived by Gorenﬂo and Mainardi, see [GorMai01], and improves the corresponding formula in the book by Samko, Kilbas and Marichev [SaKiMa93] which is not valid for ˛ D 1:

E.5.2 The Feller Fractional Integrals and Derivatives

A generalization of the Riesz fractional integral and derivative was proposed by Feller [Fel52] in a pioneering paper, recalled by Samko, Kilbas and Marichev [SaKiMa93], but only recently revised and used by Gorenﬂo and Mainardi [GorMai98]. Feller’s intention was indeed to generalize the second order space derivative appearing in the standard diffusion equation by a pseudo-differential operator whose symbol is the cumulative function (logarithm of the characteristic function) of a general Lévy stable probability distribution function according to his parametrization. Let us now show how to obtain the Feller derivative by inversion of a properly generalized Riesz potential, later called the Feller potential by Samko, Kilbas and ˛ Marichev [SaKiMa93]. Using our notation we deﬁne the Feller potential I by its symbol obtained from the Riesz potential by a suitable “rotation” by a properly restricted angle =2,i.e. ˛; if 0<˛<1; Ic˛./ Djj˛ e i.sgn /=2 ; jjÄ (E.5.12) 2 ˛;if 1<˛Ä 2; with ; 2 R. As for the Riesz potential, the case ˛ D 1 is omitted here. The ˛ integral representation of I turns out to be

˛ ˛ ˛ I .x/ D c.˛; / IC .x/ C cC.˛; / I .x/; (E.5.13) where, if 0<˛<2; ˛¤ 1;

sin Œ.˛ /=2 sin Œ.˛ C /=2 cC.˛; / D ;c.˛; / D ; (E.5.14) sin .˛/ sin.˛/ 342 E Elements of Fractional Calculus and, by passing to the limit (with D 0)

cC.2; 0/ D c.2; 0/ D1=2 : (E.5.15)

In the particular case D 0 we get

1 cC.˛; 0/ D c.˛; 0/ D ; (E.5.16) 2 cos .˛=2/ and thus, from (E.5.13)and(E.5.16) we recover the Riesz potential (E.3.1). Like the Riesz potential, the Feller potential also has the (range-restricted) semigroup property, e.g.

˛ ˇ ˛Cˇ I I D I for 0<˛<1;0<ˇ<1;˛C ˇ<1: (E.5.17)

From the Feller potential we can deﬁne by analytical continuation the Feller ˛ fractional derivative D , including also the singular case ˛ D 1;by setting

˛ ˛ D WD I ; so ( ˛; if 0<˛Ä 1; c˛ ˛ Ci.sgn /=2 D ./ Djj e ; jjÄ (E.5.18) 2 ˛;if 1<˛Ä 2:

˛ Since for D the case ˛ D 1 is included, the condition for in (E.5.18) can be shortened to

jjÄmin f˛; 2 ˛g ;0<˛Ä 2:

We note that the allowed region for the parameters ˛ and turns out to be a diamond in the plane f˛; g with vertices at the points .0; 0/, .1; 1/, .2; 0/, .1; 1/,see Fig. E.1. We call it the Feller–Takayasu diamond, in honour of Takayasu, who in his 1990 book [Tak90] ﬁrst gave the diamond representation for the Lévy-stable distributions. ˛ The representation of D .x/ can be obtained from the previous considerations. We have 8 < ˛ ˛ cC.˛; / D C c.˛; / D .x/; if ˛ ¤ 1; ˛ C D .x/ D (E.5.19) : 1 cos.=2/ D0 C sin.=2/ D .x/; if ˛ D 1:

For ˛ ¤ 1 it is sufﬁcient to note that c.˛; / D c˙.˛; / : For ˛ D 1 we need 1 O to recall the symbols of the operators D and D0 DDH , namely D D .i/ and c1 D0 Djj ; and note that E.5 The Riesz–Feller Fractional Calculus 343

Fig. E.1 The Feller–Takayasu diamond

c1 Ci.sgn /=2 D i Djj e Djj cos.=2/ .i/ sin.=2/ c1 O D cos.=2/ D0 C sin.=2/ D:

We note that in the extremal cases of ˛ D 1 we get

d D1 D˙D D˙ : (E.5.20) ˙1 dx We also note that the representation by hyper-singular integrals for 0<˛<2 (now excluding the cases f˛ D 1; ¤ 0g) can be obtained by using (E.2.18)and (E.2.21) in the first equation of (E.5.19). We get Z .1C ˛/ 1 .x C / .x/ D˛ .x/ D Œ.˛ C /=2 sin 1C˛ d 0 Z (E.5.21) 1 .x / .x/ C Œ.˛ /=2 ; sin 1C˛ d 0 which reduces to (E.5.11)for D 0: For later use we find it convenient to return to the “weight” coefficients c˙.˛; / in order to outline some properties along with some particular expressions, which can be easily obtained from (E.4.4) with the restrictions on given in (E.4.2). We obtain ( 0;if 0<˛<1; c˙ (E.5.22) Ä 0;if 1<˛Ä 2; 344 E Elements of Fractional Calculus and ( cos .=2/ >0;if 0<˛<1; cC C c D (E.5.23) cos .˛=2/ <0;if 1<˛Ä 2:

In the extremal cases we ﬁnd ( cC D 1; c D 0;if D˛; 0<˛<1; (E.5.24) cC D 0; c D 1;if DC˛; ( cC D 0; c D1;if D.2 ˛/ ; 1<˛<2; (E.5.25) cC D1; c D 0;if DC.2 ˛/ :

In view of the relation of the Feller operators in the framework of stable probability density functions, we agree to call the skewness parameter. We must note that in his original paper Feller [Fel52] used a skewness parameter ı different from our I the potential introduced by Feller is such that

˛ 2 Ic˛ D jj e i.sgn /ı ;ıD ;D ˛ı : (E.5.26) ı 2 ˛ In their book, Uchaikin and Zolotarev [UchZol99] have adopted Feller’s convention, but use the letter in place of Feller’s ı.

E.6 The Grünwald–Letnikov Fractional Calculus

Grünwald [Gru67] and Letnikov [Let68a, Let68b] independently generalized the classical deﬁnition of derivatives of integer order as limits of difference quotients to the case of fractional derivatives. Let us give an outline of their approach in a way appropriate for later use in the construction of discrete random walk models for fractional diffusion processes. For more detailed presentations, proofs of convergence, suitable function spaces etc., we refer the reader to Butzer and Westphal [ButWes00], Podlubny [Pod99], Miller and Ross [MilRos93], Samko, Kilbas and Marichev [SaKiMa93]. It is noteworthy that Oldham and Spanier in their pioneering book [OldSpa74] also made extensive use of this approach. We need one essential discrete operator, namely the shift operator Eh,whichfor h 2 R in its application to a function .x/ deﬁned in R has the effect

Eh .x/ D .x C h/ : (E.6.1)

Obviously the operators Eh,forh 2 R ; possess the group property

h1 h2 h2 h1 h1Ch2 E E D E E D E ;h1;h2 2 R ; (E.6.2) E.6 The Grünwald–Letnikov Fractional Calculus 345 and it is because of this property that we put h in the place of an exponent rather than in the place of an index. In fact, we can formally write

X1 1 Eh D e hD D hn Dn ; nŠ nD0 as a shorthand notation for the Taylor expansion of a function analytic at the point x. In the sequel, if not explicitly said otherwise, we will assume

h>0: (E.6.3)

We are now going to describe in some detail how to approximate our fractional ˛ ˛ derivatives, introduced earlier, such as DaC ;DC ;::: for ˛>0:The corresponding formulas with the minus sign in the index can then be obtained by consideration of symmetry. It is convenient to introduce the backward difference operator h as

h h D I E ; (E.6.4) whose process can, via the binomial theorem, readily be expressed in terms of the k operators Ekh D Eh :

Xn n n D .1/k Ekh ;n2 N : (E.6.5) h k kD0

0 I For complementation we also deﬁne h D : It is well-known that for sufﬁciently smooth functions and positive integer n

n n n C h h .x C ch/ D D .x/ C O.h/; h ! 0 : (E.6.6)

Here c stands for an arbitrary ﬁxed real number (independent of h). The choice c D 0 or c D n leads to completely one-sided (backward or forward, respectively) approximation, whereas the choice c D n=2 leads to approximation of second order accuracy, i.e. O.h2/ in place of O.h/:

E.6.1 The Grünwald–Letnikov Approximation in the Riemann–Liouville Fractional Calculus

The Grünwald–Letnikov approach to fractional differentiation consists in replacing in (E.6.5)and(E.6.6) the positive integer n by an arbitrary positive number ˛. Then, instead of the binomial formula, we have to use the binomial series, and we get 346 E Elements of Fractional Calculus

(formally) ! X1 ˛ ˛ D .1/k Ekh ;˛>0; (E.6.7) h k kD0 with the generalized binomial coefﬁcients ! ˛ ˛.˛ 1/:::.˛ k C 1/ .˛C 1/ D D : k kŠ .nC 1/ .˛ n C 1/

If now .x/ is a sufﬁciently well-behaved function (deﬁned in R) we have the limit relation

˛ ˛ ˛ C h h .x C ch/ ! DC .x/; h ! 0 ;˛>0: (E.6.8)

Here we usually take c D 0, but in general a fixed real number c is allowed if is smooth enough (compare Vu Kim Tuan and Gorenflo [VuGor95]). ˛ ˛ For sufficient conditions of convergence and also for convergence h h ! ˛ DC in norms of appropriate Banach spaces instead of pointwise convergence the readers may consult the above-mentioned references. From [VuGor95] one can take that in the case c D 0 convergence is pointwise (as written in (E.6.8)) and of order O.h/ if 2 C Œ˛C2.R/ and all derivatives of up to the order Œ˛ C 3 belong to L1.R/. In our later theory of random walk models for space-fractional diffusion such precise conditions are irrelevant because a posteriori we can prove their convergence. The Grünwald–Letnikov approximations will there serve us as a heuristic guide to constructional models. In fact, we will need these approximations just for 0<˛Ä 2,for0<˛<1with c D 0; for 1<˛Ä 2 with c D 1; in (E.6.9). (The case ˛ D 1 being singular in these special random walk models.) Before proceeding further, let us write in longhand notation (for the special choice c D 0) the formula (E.6.9) in the form ! X1 ˛ h˛ .1/k .x kh/ ! D˛ .x/; h ! 0C ;˛>0; (E.6.9) k C kD0 and the corresponding formula ! X1 ˛ h˛ .1/k .x C kh/ ! D˛ .x/; h ! 0C ;˛>0: (E.6.10) k kD0 N ˛ If ˛ D n 2 ,then k D 0 for k>˛D n;and we have finite sums instead of infinite series. E.6 The Grünwald–Letnikov Fractional Calculus 347

The Grünwald–Letnikov approach to the Riemann–Liouville derivative ˛ DaC .x/ where x>a>1 consists in extending .x/ by .x/ D 0 for x

˛ ˛ ˛ C GLDaC .x/ D lim h h .x/; h ! 0 ;x>a: (E.6.11) h!0C

If .x/ is smooth enough (in particular at x D a where we require .a/ D 0)we have

˛ ˛ GLDaC .x/ D DaC .x/; (E.6.12) either pointwise or in a suitable function space. Taking x a as a multiple of h;we can write the Grünwald–Letnikov derivative as a ﬁnite sum ! .xa/=h X ˛ D˛ .x/ ' h˛ ˛ .x/ D h˛ .1/k .x kh/: (E.6.13) aC h k kD0

One can show that [see Samko, Kilbas and Marichev [SaKiMa93] Eq. (20.45) and compare our formula (E.5.9)]

. C 1/ D˛ x D x˛ D D˛ x D D˛ x ; GL 0C . C 1 ˛/ 0C with x>0and >1; ˛ >0: We refrain from writing down the analogous formulas for approximation of the ˛ regressive derivative Db .x/: Let us ﬁnally point out that by formally replacing ˛ by ˛ we obtain an ˛ approximation of the operators of fractional integration as IaC with ˛>0;namely ! .xa/=h X ˛ I ˛ .x/ ' h˛ .1/k .x kh/; (E.6.14) 0C k kD0 where ! ˛ .˛/ .1/k .x kh/ D k >0: (E.6.15) k kŠ

Here we have used the convention

.˛/ kY1 k D D ˛.˛ 1/:::.˛ k C 1/ : kŠ j D0

Remark concerning notation. 348 E Elements of Fractional Calculus

In numerical analysis it is customary to write rh (instead of h) for the backward h difference operator I E andtouseh for the forward difference operator Eh I; as in Gorenﬂo [Gor97]. In our present work, however, we follow the notation employed by most workers in fractional calculus. In the modern notation the Grünwald–Letnikov fractional derivative has the form

Zt mX1 x.k/.0/t ˛Ck 1 D˛ x.t/ D C .t /m˛1x.m/. /d ; GL 0C .˛ C k C 1/ .m ˛/ kD0 0 (E.6.16) where m 1 Ä ˛

E.6.2 The Grünwald–Letnikov Approximation in the Riesz–Feller Fractional Calculus

The Grünwald–Letnikov approximation for the fractional Liouville–Weyl deriva- ˛ ˛ ˛ ˛ tives DC ;D gives us an approximation to the Riesz–Feller derivatives D0 ;D ; with 0<˛<2, according to Eqs. (E.2.7)and(E.2.19) (respectively), provided we disregard the singular case ˛ D 1.

E.7 Historical and Bibliographical Notes

Fractional calculus is the ﬁeld of mathematical analysis which deals with the investigation and application of integrals and derivatives of arbitrary order. The term fractional is a misnomer, but it has been retained following the prevailing use. The fractional calculus may be considered an old and yet novel topic. It is an old topic since, starting from some speculations of G.W. Leibniz (1695, 1697) and L. Euler (1730), it has been developed up to the present day. In fact the idea of generalizing the notion of derivative to non-integer order, in particular to the order 1/2, is contained in the correspondence of Leibniz with Bernoulli, L’Hôpital and Wallis. Euler took the ﬁrst step by observing that the result of the evaluation of the derivative of the power function has a meaning for non-integer order thanks to his Gamma function. A list of mathematicians who have provided important contributions up to the middle of the twentieth century includes P.S. Laplace (1812), J.B.J. Fourier E.7 Historical and Bibliographical Notes 349

(1822), N.H. Abel (1823–1826), J. Liouville (1832–1837), B. Riemann (1847), H. Holmgren (1865–1867), A.K. Grünwald (1867–1872), A.V. Letnikov (1868– 1872), N.Ya. Sonine (1872–1884), H. Laurent (1884), P.A. Nekrassov (1888), A. Krug (1890), J. Hadamard (1892), O. Heaviside (1892–1912), S. Pincherle (1902), G.H. Hardy and J.E. Littlewood (1917–1928), H. Weyl (1917), P. Lévy (1923), A. Marchaud (1927), H.T. Davis (1924–1936), A. Zygmund (1935–1945), E.R. Love (1938–1996), A. Erdélyi (1939–1965), H. Kober (1940), D.V. Widder (1941), M. Riesz (1949), and W. Feller (1952). In [LetChe11] A.V. Letnikov’s main results on fractional calculus are presented, including his dissertations and a long discussion between A.V. Letnikov and N.Ya. Sonine on the foundations of fractional calculus. The modern development of the ideas by Letnikov is given and their applications to underground dynamics and population dynamics are presented. However, it may be considered a novel topic as well, since it has only been the subject of specialized conferences and treatises in the last 30 years. The merit is due to B. Ross for organizing the First Conference on Fractional Calculus and its Applications at the University of New Haven in June 1974 and editing the proceedings [Ros75]. For the first monograph the merit is ascribed to K.B. Oldham and J. Spanier [OldSpa74], who, after a joint collaboration starting in 1968, published a book devoted to fractional calculus in 1974. Nowadays, to our knowledge, the list of texts in book form with a title explicitly devoted to fractional calculus (and its applications) includes around ten titles, namely Oldham and Spanier (1974) [OldSpa74], McBride (1979) [McB79], Samko, Kilbas and Marichev (1987–1993) [SaKiMa93], Nishimoto (1991) [Nis91], Miller and Ross (1993) [MilRos93], Kiryakova (1994) [Kir94], Rubin (1996) [Rub96], Podlubny (1999) [Pod99], and Kilbas, Strivastava and Trujillo (2006) [KiSrTr06]. Furthermore, we draw the reader’s attention to the treatises by Davis (1936) [Dav36], Erdélyi (1953–1954) [Bat-1, Bat-2, Bat-3], Gel’fand and Shilov (1959– 1964) [GelShi64], Djrbashian (or Dzherbashian) [Dzh66], Caputo [Cap69], Babenko [Bab86], Gorenflo and Vessella [GorVes91], West, Bologna and Grigolini (2003) [WeBoGr03], Zaslavsky (2005) [Zas05], Magin (2006) [Mag06], which contain a detailed analysis of some mathematical aspects and/or physical applications of fractional calculus. For more details on the historical development of the fractional calculus we refer the interested reader to Ross’ bibliography in [OldSpa74] and to the historical notes generally available in the above quoted texts. In recent years considerable interest in fractional calculus has been stimulated by the applications that it finds in different fields of science, including numerical analysis, economics and finance, engineering, physics, biology, etc. For economics and finance we quote the collection of articles on the topic of Fractional Differencing and Long Memory Processes, edited by Baillie and King (1996), which appeared as a special issue in the Journal of Econometrics [BaiKin96]. For engineering and physics we mention the book edited by Carpinteri and Mainardi [CarMai97], entitled Fractals and Fractional Calculus in Continuum Mechanics, which contains lecture notes of a CISM Course devoted to some 350 E Elements of Fractional Calculus applications of related techniques in mechanics, and the book edited by Hilfer (2000) [Hil00], entitled Applications of Fractional Calculus in Physics,which provides an introduction to fractional calculus for physicists, and collects review articles written by some of the leading experts. In the above books we recom- mend the introductory surveys on fractional calculus by Gorenflo and Mainardi [GorMai97] and by Butzer and Westphal [ButWes00], respectively. In addition to some books containing proceedings of international conferences and workshops on related topics, see e.g. [McBRoa85,Nis90,RuDiKi94,RuDiKi96], we mention regular journals devoted to fractional calculus, i.e. Journal of Fractional Calculus (Descartes Press, Tokyo), started in 1992, with Editor-in-Chief Prof. Nishimoto and Fractional Calculus and Applied Analysis from 1998, with Editor- in-Chief Prof. Kiryakova (Diogenes Press, Sofia). For information on this journal, please visit the WEB site www.diogenes.bg/fcaa. Furthermore, WEB sites devoted to fractional calculus have also appeared, of which we call attention to www. fracalmo.org, whose name comes from FRActional CALculus MOdelling, and the related WEB links. Appendix F Higher Transcendental Functions

F.1 Hypergeometric Functions

F.1.1 Classical Gauss Hypergeometric Functions Â Ã a; b The hypergeometric function F.aI bI cI z/ D F I z D F .aI bI cI z/ is c 2 1 deﬁned by the Gauss series

P1 .a/ .b/ ab a.a C 1/b.b C 1/ F.aI bI cI z/ D n n zn D 1 C z C z2 C ::: nD0 .c/nnŠ c c.c C 1/2Š .c/ X1 .aC n/ .b C n/ D zn .a/ .b/ .cC n/nŠ nD0 (F.1.1) on the disk jzj <1, and by analytic continuation elsewhere. In general, F.aI bI cI z/ does not exist when c D 0; 1; 2;:::. The branch obtained by introducing a cut from 1 to C1 on the real z-axis and by ﬁxing the value F.0/ D 1 is the principal branch (or principal value) of F.aI bI cI z/. For all values of c another type of classical hypergeometric function is deﬁned as

X1 .a/ .b/ 1 F.aI bI cI z/ D n n zn D F.aI bI cI z/; jzj <1; (F.1.2) .cC n/nŠ .c/ nD0 again with analytic continuation for other values of z, and with the principal branch deﬁned in a similar way. In (F.1.2) it is supposed by deﬁnition that if for an index n

© Springer-Verlag Berlin Heidelberg 2014 351 R. Gorenﬂo et al., Mittag-Lefﬂer Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2 352 F Higher Transcendental Functions we have c C n as a non-positive integer, then the corresponding term in the series is identically equal to zero (e.g. if c D1, then we omit the term with n D 0 and n D 1). This agreement is natural, because 1= .m/ D 0 for m D 0; 1; 2;:::. On the circle jzjD1, the Gauss series: (a) converges absolutely when Re .c a b/ > 0; (b) converges conditionally when 1

The principal branch of F.aI bI cI z0/ is an entire function of parameters a; b, and c for any fixed values z0; jz0j <1(see [Bat-1, p. 68]). The same is true of other branches, since the series in (F.1.2)convergesforafixedz0; jz0j <1, in any bounded domain of the complex a; b; c-space. As a multi-valued function of z, F.aI bI cI z/ is analytic everywhere except for possible branch points at z D 1,andC1.The same properties hold for F.aI bI cI z/, except that as a function of c, F.aI bI cI z/ in general has poles at c D 0; 1; 2;:::. Because of the analytic properties with respect to a; b,andc, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. For special values of parameters the hypergeometric function coincides with elementary functions

ln.1 z/ .a/ F.1I 1I 2I z/ D ; z Â Ã 1 1 C z .b/ F.1=2I 1I 3=2I z2/ D ln ; 2z 1 z

arctan z .c/ F.1=2I 1I 3=2Iz2/ D ; z

arcsin z .d/ F.1=2I 1=2I 3=2I z2/ D ; z p Á ln z C 1 C z2 .e/ F.1=2I 1=2I 3=2Iz2/ D ; z

.f / F.aI bI bI z/ D .1 z/a;

1 .g/ F.aI 1=2 C aI 1=2I z2/ D .1 C z/2a C .1 z/2a 2 (see also the additional formulas below in Sect. F.5,cf.[NIST]). F.1 Hypergeometric Functions 353

We have the following asymptotic formulas for the hypergeometric function near the branch point z D 1

F.aI bI a C bI z/ .aC b/ .i/ lim D I z!10 ln .1 z/ .a/ .b/ Â Ã .c/ .c a b/ .ii/ lim .1 z/aCbc F.aI bI cI z/ z!10 .c a/ .c b/ .c/ .aC b c/ D ; where Re .c a b/ D 0; c 6D a C bI .a/ .b/ .iii/ lim .1 z/aCbc F.aI bI cI z/ z!10 .c/ .aC b c/ D ; where Re .c a b/ < 0: .a/ .b/

All these formulas can be immediately veriﬁed, they follow from the deﬁnition (F.1.1).

F.1.2 Euler Integral Representation: Mellin–Barnes Integral Representation

If Re c>Re b>0, the hypergeometric function satisﬁes the Euler integral representation (cf., [Bat-1, p. 59])

Z1 .c/ t b1.1 t/cb1 F.aI bI cI z/ D dt: (F.1.3) .b/ .c b/ .1 zt/a 0

Here the right-hand side is a single-valued analytic function of z in the domain jarg .1 z/j <. Therefore, this formula determines an analytic continuation of the hypergeometric function F.aI bI cI z/ into this domain too. In order to prove (F.1.3) in the unit disk jzj <1it is sufﬁcient to expand the function .1 zt/a into a binomial series and calculate the series term-by-term by using standard formulas for the Beta function. The Euler integral representation can be rewritten as

.1ZC/ i .c/ei.bc/ t b1.1 t/cb1 F.aI bI cI z/ D dt; .b/ .c b/2sin .c b/ .1 zt/a 0 Re b>0;jarg .1 z/j <;c b 6D 1;2;:::I 354 F Higher Transcendental Functions

Z1 i .c/eib t b1.1 t/cb1 F.aI bI cI z/ D dt; .b/ .c b/2sin b .1 zt/a .0C/ Re c>Re b>0;jarg .z/j <;b6D 1;2;:::I F.aI bI cI z/ Z i .c/eic t b1.1 t/cb1 D dt; .b/ .c b/4sin b sin .c b/ .1 zt/a .1C;0C;1;0/ jarg .z/j

In each case we suppose that the path of integration starts and ends at corresponding points on the Riemann surface of the function t b1.1t/cb1.1zt/a with real t, 0 Ä t Ä 1,wheret b , .1 t/cb mean the principal branches of these functions and .1 zt/a is deﬁned in such a way that .1 zt/a ! 1 whenever z ! 0. The second type of integral representation is the so called Mellin–Barnes representation (see Appendix D)

CZi1 .a/ .b/ 1 .aC s/ .b C s/ .s/ F.aI bI cI z/ D .z/s ds; .c/ 2i .cC s/ i1 (F.1.4) where jarg .z/j

F.1.3 Basic Properties of Hypergeometric Functions

In the domain of deﬁnition the hypergeometric function satisﬁes some differential relations

d ab dz F.aI bI cI z/ D c F.aC 1I b C 1I c C 1I z/;

n d .a/n.b/n n F.aI bI cI z/ D F.aC nI b C nI c C nI z/; dz .c/n Á (F.1.5) n c1 cn1 d z F.aI bI cI z/ D .cn/nz F.a nI b nI c nI z/; dzn .1z/cab .1z/cCnab Á n d a1 aCn1 z dz z z F.aI bI cI z/ D .a/nz F.aC nI bI cI z/: F.1 Hypergeometric Functions 355 Á Á n n d d Here the operator z dz z is deﬁned by the operator identity z dz z ./ D n dn n z dzn z ./; n D 1;2;:::.

F.1.4 The Hypergeometric Differential Equation

If the second order homogeneous differential equation has at most three singular points, we can assume that these are the points 0; 1; 1. If all these points are regular (see [Bol90]), then the equation can be reduced to the form

d2w dw z.z 1/ C Œc .a C b C 1/z abw D 0: (F.1.6) dz2 dz

This is the hypergeometric differential equation. It has regular singularities at z D 0; 1; 1, with corresponding exponent pairs f0; 1 cg, f0; c a bg, fa; bg, respectively. Here we use the standard terminology for the complex ordinary differential equation (see, e.g., [NIST, §2.7(i)])

d2w dw C f.z/ C g.z/w D 0: dz2 dz

A point z0 is an ordinary point for this equation if the coefficients f and g are analytic in a neighbourhood of z0. In this case all solutions of the equation are analytic in a neighbourhood of z0. All other points z0 are called singular points (or simply, singularities) for the differential equation. If both .z z0/f .z/ and 2 .z z0/ g.z/ are analytic in a neighbourhood of z0, then this point is a regular singularity. All other singularities are called irregular. An irregular singularity z0 is l 2l of rank l 1 if l is the least integer such that .z z0/ f.z/ and .z z0/ g.z/ are analytic in a neighbourhood of z0. The most common type of irregular singularity for special functions has rank 1 and is located at infinity. In this case the coefficients f , g have series representations

X1 f X1 g f.z/ D s ;g.z/ D s ; zs zs sD0 sD0 where at least one of f0;g0;g1 is non-zero. Regular singularities are characterized by a pair of exponents (or indices) that are roots ˛1;˛2 of the indicial equation

Q.˛/ Á ˛.˛ 1/ C f0˛ C g0 D 0;

2 where f0 D limz!z0 .z z0/f .z/ and g0 D limz!z0 .z z0/ g.z/. Provided that .˛1 ˛2/ 62 Z the differential equation has two linear independent solutions 356 F Higher Transcendental Functions

X1 ˛j s wj D .z z0/ as;j .z z0/ ;jD 1; 2: sD0

In the case of the hypergeometric differential equation, when none of c;c a b; a b is an integer, we have the pair f1.z/, f2.z/ of fundamental solutions. They are also numerically satisfactory ([NIST, §2.7(iv)]) in a neighbourhood of the corresponding singularity.

F.1.5 Kummer’s and Tricomi’s Conﬂuent Hypergeometric Functions

Kummer’s differential equation

d2w dw z C .c z/ aw D 0 (F.1.7) dz2 dz has a regular singularity at the origin with indices 0 and 1 b, and an irregular singularity at infinity of rank one. It can be regarded as the limiting form of the hypergeometric differential equation (F.1.6) that is obtained on replacing z by z=b, letting b !1, and subsequently replacing the symbol c by b. In effect, the regular singularities of the hypergeometric differential equation at b and 1 coalesce into an irregular singularity at 1. Equation (F.1.7) is called the confluent hypergeometric equation, and its solutions are called confluent hypergeometric functions. Two standard solutions to Eq. (F.1.7) are the following:

X1 .a/ a a.a C 1/ M.aI cI z/ D n zn D 1 C z C z2 C ::: (F.1.8) .c/ nŠ c c.c C 1/2Š nD0 n and

X1 .a/ 1 a a.a C 1/ M.aI cI z/ D n zn D C z C z2 C :::: .cC n/nŠ .c/ .cC 1/ .cC 2/2Š nD0 (F.1.9)

The ﬁrst of these functions M.aI cI z/ does not exist if c is a non-positive integer. For all other values of parameters the following identity holds:

M.aI cI z/ D .c/M.aI cI z/:

The series (F.1.8)and(F.1.9) converge for all z 2 C. M.aI cI z/ is an entire function in z and a, and is meromorphic in c. M.aI cI z/ is an entire function in z;a, and c. F.1 Hypergeometric Functions 357

The function M.aI cI z/ is known as the Kummer conﬂuent hypergeometric function. Sometimes the notation

M.aI cI z/ D 1F1.aI cI z/ is used, which is also Humbert’s symbol [Bat-1, p. 248]

M.aI cI z/ D ˚.aI cI z/:

Another standard solution to the conﬂuent hypergeometric equation (F.1.7)isthe function U.aI cI z/ which is determined uniquely by the property

U.aI cI z/ za; z !1; jarg zj < ı: (F.1.10)

Here ı is an arbitrary small positive constant. In general, U.aI cI z/ has a branch point at z D 0. The principal branch corresponds to the principal value of za in (F.9), and has a cut in the z-plane along the interval .1I 0. The function U.aI cI z/ was introduced by Tricomi (see, e.g., [Bat-1, p. 257]). Sometimes it is called the Tricomi conﬂuent hypergeometric function. It is related to the Erdélyi function 2F0.aI cI z/

a 2F0.aI cI1=z/ D z U.aI a c C 1I z/ and the Kummer conﬂuent hypergeometric function

.1 c/ .c 1/ U.aI cI z/ D M.aI cI z/ C z1aM.a c C 1I 2 cI z/: .a c C 1/ .a/

The following notation (Humbert’s symbol) is used too (see, e.g., [Bat-1,p.255]):

U.aI cI z/ D .aI cI z/:

Integral Representations

Two main types of integral representations for conﬂuent hypergeometric functions can be mentioned here. First of all these are representations analogous to the Euler integral representation of the classical hypergeometric function. The integral representation of Kummer’s conﬂuent hypergeometric function

Z1 .c/ M.aI cI z/ D etzt a1.1 t/ca1dt; Re c>Re a>0; .a/ .c a/ 0 (F.1.11) can be immediately veriﬁed by expanding the exponential function etz. 358 F Higher Transcendental Functions

The formula

Z1 1 etzt a1.1 C t/ca1dt; Re a>0; (F.1.12) .a/ 0 gives the solution of the differential equation (F.1.7) in the right half-plane Re z >0 and coincides in this domain with U.aI cI z/. The analytic continuation of (F.1.12) yields the integral representation of Tricomi’s conﬂuent hypergeometric function U.aI cI z/. Another type of integral representation uses Mellin–Barnes integrals. For conﬂu- ent hypergeometric functions this type of integral representation has the form

ZCi1 1 .c/ .s/ .a C s/ M.aI cI z/ D .z/sds; (F.1.13) 2i .a/ .cC s/ i1 jarg .z/j <=2; Re a<<0;c6D 0; 1; 2; : : : I

ZCi1 1 .s/ .a C s/ .1 c s/ U.aI cI z/ D .z/s ds; (F.1.14) 2i .cC s/ .a c C 1/ i1 jarg .z/j <3=2; Re a<

The conditions on the parameters in (F.1.13)andin(F.1.14) can be relaxed by suitable deformation of the contour of integration. Thus, (F.1.13) is valid with any whenever a is not a non-negative integer, provided that the contour of integration separates the poles of .s/ from the poles of .aC s/. Similarly, (F.1.14) is valid with any whenever neither a nor a c C 1 is a non-negative integer, provided that the contour of integration separates the poles of .s/ .1 c s/ from the poles of .aC s/. The conditions on arg z cannot be relaxed.

Asymptotics

As z !1then (see, e.g., [NIST, p. 328])

ezzac X1 .1 a/ .c a/ M.aI cI z/ n n zn .a/ nŠ nD0 1 e˙iaza X .a/ .a c C 1/ C n n .z/n; (F.1.15) .c a/ nŠ nD0 =2 C ı<˙arg z <3=2 ı; a 6D 0; 1; 2;:::I c a 6D 0; 1; 2;:::I F.1 Hypergeometric Functions 359

X1 .a/ .a c C 1/ U.aI cI z/ za n n .z/n; (F.1.16) nŠ nD0 jarg zj <3=2 ı; with ı being sufﬁciently small positive:

Other asymptotic formulas with respect to large values of the variable z and/or with respect to large values of the parameters a and c can be found in [NIST, pp. 330–331].

Relation to Elementary and Other Special Functions

In the following special cases the conﬂuent hypergeometric functions coincide with elementary functions:

M.aI aI z/ D ezI ez M.1I 2I 2z/ D sinh zI z M.0I cI z/ D U.0I cI z/ D 1I U.aI a C 1I z/ D zaI with incomplete gamma function .a;z/ (in the case when a c is an integer or a is a positive integer):

M.aI a C 1Iz/ D ezM.1I a C 1I z/ D aza.a;z/I with the error functions erf and erfc: p M.1=2I 3=2Iz2/ D erf.z/I 2z p 2 U.1=2I 1=2I z2/ D ez erfc.z/I with the orthogonal polynomials:

– With the Hermite polynomials H.z/:

nŠ M.nI 1=2I z2/ D .1/n H .z/I .2n/Š 2n nŠ M.nI 3=2I z2/ D .1/n H .z/I .2n C 1/Š2z 2nC1 2n U.1=2 n=2I 3=2I z2/ D H .z/I z n 360 F Higher Transcendental Functions

.˛/ – With the Laguerre polynomials L .z/:

n n .˛/ U.nI ˛ C 1I z/ D .1/ .˛ C 1/nM.nI ˛ C 1I z/ D .1/ nŠLLn .z/I

with the modiﬁed Bessel functions I;K (in the case when c D 2b):

z M. C 1=2I 2 C 1I z/ D .C 1/e .z=2/ I.z/I

1 z U. C 1=2I 2 C 1I z/ D p e .2z/ K.z/:

F.1.6 Generalized Hypergeometric Functions and Their Properties Â Ã a Generalized hypergeometric functions F (or F I z D F .a; bI z/ D p q p q b p q pFq.a1;:::;apI b1;:::;bq I z/) are deﬁned by the following series (where none of the parameters bj is a non-positive integer): Â Ã X1 n a1;a2;:::ap .a1/n.a2/n :::.ap/n z pFq I z D : (F.1.17) b1;b2;:::bq .b / .b / :::.b / nŠ nD0 1 n 2 n q n This series converges for all z whenever p0, convergent except at z D 1 if 1 q C1. However, when one or more of the top parameters aj is a non-positive integer the series terminates and the generalized hypergeometric function is a polynomial in z. Note that if m D maxfa1;:::;aq g is a positive integer, then the following identity holds: Â Ã Â Ã m pCqC1 m; a .a/m.z/ m; 1 m b .1/ pC1Fq I z D qC1Fp I ; b .b/m 1 m a z which can be used to interchange a and b. F.2 Wright Functions 361

If p Ä q D 1 and z is not a branch point of the generalized hypergeometric function pFq , then the function Â Ã Â Ã X1 n a 1 a .a1/n :::.ap /n z F I z D F I z D p q b .b /::: .b / p q b .b C n/::: .b C n/ nŠ 1 q nD0 1 q is an entire function of each parameter a1;:::;ap;b1;:::;bq.

F.2 Wright Functions

F.2.1 The Classical Wright Function

In this section we present some deﬁnitions and properties of the so-called Wright functions given in [Bat-1, Section 4.1], [Bat-3, Section 18.1], [KiSrTr06, Section 1.11], and [Mai10, Appendix F]. The simplest Wright function .˛;ˇI z/ is deﬁned for z;˛;ˇ 2 C by the series 2 3 ˇ ˇ X1 k 4 ˇ 5 1 z .˛;ˇI z/ D 01 ˇ z WD : (F.2.1) .˛kC ˇ/ kŠ .ˇ; ˛/ kD0

If ˛>1, this series is absolutely convergent for all z 2 C, while for ˛ D1 it is absolutely convergent for jzj <1and for jzjD1 and Re.ˇ/ > 1;see [KiSrTr06, Sec. 1.11]. Moreover, for ˛>1, .˛;ˇI z/ is an entire function of z. Using formulas for the order (B.2.3) and the type (B.2.4) of an entire function and Stirling’s asymptotic formula for the Gamma function (A.1.24) one can deduce 1 that for ˛>1 the Wright function .˛;ˇI z/ has order D and type ˛ C 1 1 1 D .˛ C 1/˛ .˛ C 1/ D ˛. The function .˛;ˇI z/ was introduced by Wright [Wri33],whoin[Wri34]and [Wri40b] investigated its asymptotic behaviour at inﬁnity provided that ˛>1. When ˛ D 1 and ˇ D C 1, the function .1; C 1I˙z2=4/ is expressed in terms of the Bessel functions J.z/ and I.z/, given in Appendix G by formulas (G.1) and (G.10): Â Ã Â Ã Â Ã Â Ã z2 2 z2 2 1; C 1I D J .z/; 1; C 1I D I .z/: (F.2.2) 4 z 4 z 362 F Higher Transcendental Functions

F.2.2 Mellin–Barnes Integral Representation and Asymptotics

The integral representation of (F.2.1) in terms of the Mellin–Barnes contour integral is given by Z 1 .s/ .˛;ˇI z/ D .z/s ds; (F.2.3) 2i L .ˇ ˛s/ where the path of integration L separates all the poles at s Dk.k2 N0/ to the left. If L D . i1; C i1/. 2 R/, then the representation (F.2.3) is valid if either of the following conditions holds:

.1 ˛/ 0<˛<1; jarg.z/j < ; z ¤ 0 (F.2.4) 2 or

˛ D 1; Re.ˇ/ > 1 C 2; arg.z/ D 0; z ¤ 0: (F.2.5)

The relation (F.2.3) means that the Mellin transform of (F.2.1)isgivenbythe formula .s/ MŒ .˛; ˇI t/.s/ D .Re.s/ > 0/: (F.2.6) .ˇ ˛s/

The Laplace transform of (F.2.1) is expressed in terms of the Mittag-Lefﬂer function: Â Ã 1 1 LŒ .˛; ˇI t/.s/ D E .˛ > 1I ˇ 2 CI Re.s/ > 0/; (F.2.7) s ˛;ˇ s see [Bat-3, Section 18.2]. If z 2 C and jarg.z/jÄ .0<

.˛;ˇI z/ ÄÂ Ã " Â Ã !# 1 1 1=.1C˛/ D a .˛z/.1ˇ/=.1C˛/exp 1 C .˛z/1=.1C˛/ 1 C O ; 0 ˛ z (F.2.8)

1=2 where a0 D Œ2.˛ C 1/ .z !1/. The following differentiation formula for .˛;ˇI z/ is a direct consequence of the deﬁnition (F.2.1): Â Ã d n .˛;ˇI z/ D .˛;˛ C nˇI z/.n2 N/: (F.2.9) dz F.2 Wright Functions 363

F.2.3 The Bessel–Wright Function: Generalized Wright Functions and Fox–Wright Functions

When ˛ D , ˇ D C1,andz is replaced by z, the function .˛;ˇI z/ is denoted by J .z/:

X1 1 .z/k J .z/ WD .; C 1Iz/ D ; (F.2.10) .kC C 1/ kŠ kD0 and this function is known as the Bessel–Wright function, or the Wright generalized Bessel function,1 see [PrBrMa-V2]and[Kir94, p. 352]. When D 1, the Bessel function of the ﬁrst kind (G.1) is connected with (F.2.10)by Á Â Ã z z2 J .z/ D J 1 : (F.2.11) 2 4

p If D q is a rational number then [Kir94, p. 352] the Bessel–Wright function satisﬁes the following differential equation of order .p C q/ 2 3 Â Ã .z/q Y 1 d 4 p C q z d 5 J .z/ D 0; (F.2.12) qqpp q dz j j D1 where ( j q ;1Ä j Ä q 1; dj D j CC1q 1 q ;qÄ j Ä q C p:

 The original differential equation for J .z/, equivalent to (F.2.12), was formu- lated by Wright [Wri33]: Â Ã Â Ã p q d Cp d .1/qz=J .z/ D z11= z J .z/: dz dz 

There exists a further generalization of the Bessel–Wright function (the so-called generalized Bessel–Wright function) given in [Pat66, Pat67]: Á 1 2k z C2 X .1/k z J .z/ D 2 ;>0: (F.2.13) ; 2 .C k C C 1/ . C k C 1/ kD0

1Also misnamed the Bessel–Maitland function after E.M. Wright’s second name Maitland. 364 F Higher Transcendental Functions

This function generates the Lommel function, and is a special case of the four- parametric Mittag-Lefﬂer function. The more general function pq .z/ is deﬁned for z 2 C, complex al ;bj 2 C,and real ˛l , ˇj 2 R .l D 1; ;pI j D 1; ;q/by the series 2 3 ˇ Q .a ;˛ / 1 p l l 1;p ˇ X .a C ˛ k/ k 4 ˇ 5 Q lD1 l l z pq .z/ D pq ˇ z WD q (F.2.14) j D1 .bj C ˇj k/ kŠ .bl ;ˇl /1;q kD0

.z;al ;bj 2 CI ˛l ;ˇj 2 RI l D 1; ;pI j D 1; ;q/:

This general (Wright or, more appropriately, Fox–Wright) function was investigated by Fox [Fox28] and Wright [Wri35b,Wri40b,Wri40c], who presented its asymptotic expansion for large values of the argument z under the condition

Xq Xp ˇj ˛l > 1: (F.2.15) j D1 lD1

If these conditions are satisﬁed, the series in (F.2.14) is convergent for any z 2 C. This result follows from the assertion [KiSrTr06, Theor. 1.]:

Theorem F.1. Let al ;bj 2 C and ˛l , ˇj 2 R .l D 1; pI j D 1; ;q/and let

Xq Xp D ˇj ˛l ; (F.2.16) j D1 lD1 Yp Yq ˛l ˇj ı D j˛l j jˇj j ; (F.2.17) lD1 j D1 and

q p X X p q D b a C : (F.2.18) j l 2 j D1 lD1

(a) If >1, then the series in (F.2.14) is absolutely convergent for all z 2 C. (b) If D1, then the series in (F.2.14) is absolutely convergent for jzj <ıand for jzjDı and Re./ > 1=2.

When ˛l ;ˇj 2 R .l D 1; ;pI j D 1; ;q/, the generalized Wright function pq.z/ has the following integral representation as a Mellin–Barnes contour integral: " # Q ˇ Z p .al ;˛l /1;p ˇ 1 .s/ .a ˛ s/ ˇ Q lD1 l l s pq ˇ z D q .z/ ds; (F.2.19) 2i C .bl ;ˇl /1;q j D1 .bj ˇj s/ F.3 Meijer G-Functions 365 where the path of integration C separates all the poles at s Dk.k2 N0/ to the left and all the poles s D .al C nl /=˛l .l D 1; ;pI nl 2 N/ to the right. If C D . i1;C i1/. 2 R), then representation (F.2.19) is valid if either of the following conditions holds:

.1 / <1; jarg.z/j < ; z ¤ 0 (F.2.20) 2 or 1 D 1; . C 1/ C < Re./; arg.z/ D 0; z ¤ 0: (F.2.21) 2 Conditions for the representation (F.2.19) were also given for the case when C D L1 .L1/ is a loop situated in a horizontal strip starting at the point 1 C i'1 .1Ci'1/ and terminating at the point 1 C i'2 .1Ci'2/ with 1 <'1 < '2 < 1. If we put ˛l D 1; 1 Ä l Ä p,andˇl D 1; 1 Ä l Ä q, in representation (F.2.14) then we get the following relation of the generalized Wright function pq with the generalized hypergeometric function pFq : " # Q ˇ p .al ;1/1;p ˇ .a / ˇ QlD1 l pq ˇ z D q pFq .a1;:::;ap I b1;:::;bqI z/: (F.2.22) .bl ;1/1;q lD1 .bl /

F.3 Meijer G-Functions

F.3.1 Deﬁnition via Integrals: Existence

In [Mei36] more general classes of transcendental functions were introduced via generalization of the Gauss hypergeometric functions presented in the form of series (commonly known now as Meijer G-functions): Ä ˇ ˇ m;n ˇ a1;:::;ap Gp;q z ˇ : (F.3.1) b1;:::;bq

This deﬁnition is due to the relation between the generalized hypergeometric function and the Meijer G-functions (see, e.g., [Sla66, p. 42]): Ä ˇ ˇ m;n ˇ 1 a1;:::;1 ap .a1/::: .ap/ Gp;q z ˇ D pFq .aI bI z/: (F.3.2) 0; 1 b1;:::;1 bq .b1/::: .bq/ 366 F Higher Transcendental Functions

Later this deﬁnition was replaced by the Mellin–Barnes representation of the G-function Ä ˇ Z ˇ 1 m;n ˇ a1;:::;ap Gm;n s Gp;q z ˇ D p;q .s/z ds; (F.3.3) b1;:::;bq 2i L where L is a suitably chosen path, z 6D 0, zs WD expŒs.ln jzjCiargz/ with a single valued branch of arg z, and the integrand is deﬁned as Q Q m .b s/ n .1 a C s/ Gm;n Q kD1 k j DQ1 j p;q .s/ D q p : (F.3.4) kDmC1 .1 bk C s/ j DnC1 .aj s/

In (F.3.4) the empty product is assumed to be equal to 1, parameters m; n; p; q satisfy the relation 0 Ä n Ä q, 0 Ä n Ä p, and the complex numbers aj , bk are such that no pole of .bk s/;k D 1;:::;m, coincides with a pole of .1aj Cs/;j D 1;:::;n.

F.3.2 Basic Properties of the Meijer G-Functions

10. Symmetry. The G-functions are symmetric with respect to their parameters in the following sense: if the value of a parameter from the group .a1;:::;an/ (respectively, from the group .anC1;:::;ap/) is equal to the value of a parameter from the group .bmC1;:::;bq/ (respectively, to the value of a parameter from the group .b1;:::;bm/), then these parameters can be excluded from the deﬁnition of the G- function, i.e. the “order” of the G-function decreases. For instance, if a1 D bq, then Ä ˇ Ä ˇ ˇ ˇ m;n ˇ a1;:::;ap m;n1 ˇ a2;:::;ap Gp;q z ˇ D Gp1;q1 z ˇ : (F.3.5) b1;:::;bq b1;:::;bq1

20. Shift of parameters. The two properties below indicate how to “shift” parameters. They follow from the change of variable in the deﬁnition of the G-functions (F.3.3). Ä ˇ Ä ˇ ˇ ˇ m;n ˇ a1;:::;ap m;n ˇ a1 C ;:::;ap C z Gp;q z ˇ D Gp;q z ˇ I (F.3.6) b1;:::;bq b1 C ;:::;bq C Ä ˇ Ä ˇ ˇ ˇ m;n ˇ a1;:::;ap m;n 1 ˇ 1 b1;:::;1 bq Gp;q z ˇ D Gp;q ˇ : (F.3.7) b1;:::;bq z 1 a1;:::;1 ap

30. Multiplication formula. F.3 Meijer G-Functions 367

For any natural number r 2 N the following formula is valid: Ä ˇ ˇ m;n ˇ a1;:::;ap Gp;q z ˇ b1;:::;bq Ä ˇ zr ˇ c ;:::;c ;:::;c ;:::;c D .2/urvGmr;nr ˇ 1;1 1;r p;1 p;r ; pr;qr r.qp/ ˇ (F.3.8) r d1;1;:::;d1;r ;:::;dq;1;:::;dq;r where

a C l 1 b C l 1 c D j ;.lD 1;:::;r/I d D k ;.lD 1;:::;r/: j;l r k;l r

F.3.3 Special Cases

The basic elementary functions can be represented as G-function with special values of parameters. They mostly follow from the relation between the generalized hypergeometric function pFq and the Meijer G-function (F.3.2). Let us recall some of these representations.

0;1 exp z D G1;0 Œz j0I (F.3.9) Ä ˇ 2 ˇ p 1;0 z ˇ1 sin z D G ˇ ;0 I (F.3.10) 0;2 4 2 Ä ˇ 2 ˇ p 1;0 z ˇ 1 cos z D G ˇ0; I (F.3.11) 0;2 4 2 Ä ˇ ˇ 1;2 ˇ 1; 1 log .1 C z/ D G z ˇ .jzj <1/I (F.3.12) 2;2 1; 0 Ä ˇ ˇ 1 1;2 2 ˇ 1; 1 arcsin z D p G2;2 z ˇ 1 .jzj <1/I (F.3.13) 2 2 ;0 Ä ˇ 1 ˇ 1 1;2 2 ˇ 1; 2 arctan z D G2;2 z ˇ 1 .jzj <1/I (F.3.14) 2 2 ;0 Ä ˇ ˇ 1;0 ˇ C 1 z D G z ˇ .jzj <1/I (F.3.15) 1;1 Ä ˇ ˇ ˛ 1 1;1 ˇ 1 ˛ .1 ˙ z/ D G z ˇ .jzj <1;Re ˛>0/I (F.3.16) .˛/ 1;1 0 Ä ˇ ˇ ˛ 1;0 ˇ 1 ˛ .1 ˙ z/ D .1 ˛/G z ˇ .jzj <1;Re ˛<1/: (F.3.17) 1;1 0 368 F Higher Transcendental Functions

F.3.4 Relations to Fractional Calculus

Let us consider the Riemann–Liouville fractional integral (see (E.1.3)) Z x ˛ 1 ˛1 IaC .x/ WD .x / ./d; a0: .˛/ a

It is well-known (see, e.g. [SaKiMa93, Ch. 4]) that this integral can be extended into the complex domain. For this we ﬁx a point z 2 C and introduce the multi-valued function

.z /˛1:

By ﬁxing an arbitrary single-valued branch of this function in the complex plane cut along the line L starting at D a and ending at D1and containing the point D z we deﬁne the Riemann–Liouville fractional integral in the complex domain

Zz 1 I ˛ f.z/ D .z /˛1 f./d; (F.3.18) aC .˛/ a where the integration is performed along a part of the above-described contour (cut) L starting at D a and ending at D z. Analogously, the so-called “right- sided” Riemann–Liouville fractional integral (or Weyl type fractional integral) in the complex domain is deﬁned by

Z1 1 I ˛ f.z/ D .z /˛1 f./d: (F.3.19) .˛/ z

The density f in both formulas (F.3.18)and(F.3.19)isassumedtobedeﬁnedin a neighbourhood of the contour L. If this function is analytic in the whole plane, then by the Cauchy Integral Theorem one can replace the contour L by the straight interval Œa; z andbytherayŒz; 1/, respectively. Let us present the formulas which show that the Riemann–Liouville fractional integration of the Meijer G-function yields the Meijer G-function with other values of parameters (see, e.g. [Kir94, p. 318], [PrBrMa-V3]).

Zz Ä ˇ ˇ ˛ m;n 1 ˛1 m;n ˇ a1;:::;ap IaC Gp;q .z/ D .z / Gp;q ˇ d .˛/ b1;:::;bq a Z1 Ä ˇ ˛1 ˇ ˛ .1 / m;n ˇ a1;:::;ap D z Gp;q z ˇ d .˛/ b1;:::;bq 0 F.3 Meijer G-Functions 369 Ä ˇ ˇ ˛ m;nC1 ˇ 0; a1;:::;ap D z GpC1;qC1 ˇ : (F.3.20) b1;:::;bq; ˛

This formula is valid when Re bk > 1; k D 1;:::;m,andp Ä q.Ifp D q, then it is valid for jzj <1Á .Forp C q<2.mC n/ we require additionally that pCq jarg zj < m C n 2 .

Z1 Ä ˇ ˇ ˛ m;n 1 ˛1 m;n ˇ a1;:::;ap I Gp;q .z/ D .z / Gp;q ˇ d .˛/ b1;:::;bq z Z1 Ä ˇ ˛1 ˇ ˛ . 1/ m;n ˇ a1;:::;ap D z Gp;q z ˇ d .˛/ b1;:::;bq 1 Ä ˇ ˇ ˛ mC1;n ˇ a1;:::;ap;0 D z GpC1;qC1 ˇ : (F.3.21) ˛; b1;:::;bq

This formula is valid when 01.Forp C q<2.mC n/ we require additionally pCq that jarg zj < m C n 2 .

F.3.5 Integral Transforms of G-Functions

The structure of the Meijer G-function is very similar to that of the Mellin integral transform. Hence, the Mellin transform of the Meijer G-function is up to a power factor equal to the ratio of the products of -functions (see, e.g., [BatErd54b, p. 301], [Kir94,p.318]).

Z1 Ä ˇ Á ˇ M m;n s1 m;n ˇ a1;:::;ap Gp;q .t/ .s/ D t Gp;q t ˇ dt b1;:::;bq 0 Qm Qn .bk C s/ .1 aj s/ sGm;n s kD1 j D1 D p;q .s/ D Qq Qp ; (F.3.22) .1 bk s/ .aj C s/ kDmC1 j DnC1 1 where, e.g., p Cq<2.mCn/; jarg j < m C n .p C q/ , min Re bk < 2 1ÄkÄm Re <1 max Re aj . Other conditions under which formula (F.3.22) is valid 1Äj Än can be found in [Luk1, pp. 157–159] and [BatErd54b, pp. 300–301]. We also mention several formulas for the Mellin integral transform presented in [BatErd54a, pp. 295–296]. 370 F Higher Transcendental Functions

The Laplace transform of the Meijer G-function can be determined by the relation:

Z1 Ä ˇ Á ˇ L m;n tx m;n ˇ a1;:::;ap Gp;q .x/ .t/ D e Gp;q x ˇ dx b1;:::;bq 0 Ä ˇ ˇ 1 m;nC1 ˇ 0; a1;:::;ap D t GpC1;q ˇ ; (F.3.23) t b1;:::;bq 1 where, e.g., p C q<2.mC n/; jarg j < m C n 2 .p C q/ , jarg tj <=2, Re bk 1; k D 1;:::;m. More general formulas for the Laplace transform of the Meijer G-function with different weights can also be found in [Luk1, pp. 166–169] and [BatErd54b, p. 302].

F.4 Fox H -Functions

F.4.1 Deﬁnition via Integrals: Existence

A straightforward generalization of the Meijer G-functions are the so-called Fox H-functions, introduced and studied by Fox in [Fox61]. According to a standard notation the Fox H-functions are deﬁned by Ä ˇ Z ˇ 1 m;n m;n ˇ .a1;˛1/;:::;.ap ;˛p/ Hm;n s Hp;q .z/ D Hp;q z ˇ D p;q .s/ z ds; .b1;ˇ1/;:::;.bq;ˇq / 2i L (F.4.1) where L is a suitable path in the complex plane C to be found later on, zs D expfs.log jzjCi argz/g,and

A.s/B.s/ Hm;n.s/ D ; (F.4.2) p;q C.s/D.s/ Ym Yn A.s/ D .bj ˇj s/; B.s/ D .1 aj C ˛j s/; (F.4.3) j D1 j D1 Yq Yp C.s/ D .1 bj C ˇj s/; D.s/ D .aj ˛j s/; (F.4.4) j DmC1 j DnC1

C with 0 Ä n Ä p, 1 Ä m Ä q, faj ;bj g2C, f˛j ;ˇj g2R .Asusual,anempty product, when it occurs, is taken to be equal to 1. Hence

n D 0 , B.s/ D 1; mD q , C.s/ D 1; nD p , D.s/ D 1: F.4 Fox H -Functions 371

Due to the occurrence of the factor zs in the integrand of (D.1.1), the H-function is, in general, multi-valued, but it can be made one-valued on the Riemann surface of log z by choosing a proper branch. We also note that when the ˛sandˇs are equal m;n to 1, we obtain the Meijer G-functions Gp;q .z/, thus the Meijer G-functions can be considered as special cases of the Fox H-functions: Ä ˇ Ä ˇ ˇ ˇ m;n ˇ .a1;1/;:::;.ap ;1/ m;n ˇ a1;:::;ap Hp;q z ˇ D Gp;q z ˇ : .b1;1/;:::;.bq ;1/ b1;:::;bq

The above integral representation of H-functions in terms of products and ratios of Gamma functions is known to be of Mellin–Barnes integral type. A compact notation is usually adopted for (F.4.1): Ä ˇ ˇ m;n m;n ˇ.aj ;˛j /j D1;:::;p Hp;q .z/ D Hp;q z ˇ : (F.4.5) .bj ;ˇj /j D1;:::;q

The singular points of the kernel H are the poles of the Gamma functions appearing in the expressions of A.s/ and B.s/, that we assume do not coincide. Denoting by P.A/ and P.B/ the sets of these poles, we write P.A/ \ P.B/ D;. The conditions for the existence of the H-functions can be determined by inspecting the convergence of the integral (F.4.1), which can depend on the selection of the contour L and on certain relations between the parameters fai ;˛i g (i D 1;:::;p) and fbj ;ˇj g (j D 1;:::;q). For the analysis of the general case we refer to the specialized treatises on H-functions, e.g., [MatSax73, MaSaHa10, SrGuGo82] and, in particular to the paper by Braaksma [Bra62], where an exhaustive discussion on the asymptotic expansions and analytical continuation of these functions can be found, see also [KilSai99]. In the following we limit ourselves to recalling the essential properties of the H-functions, preferring to analyze later in detail those functions related to fractional diffusion. As will be shown later, this phenomenon depends on one real independent variable and three parameters; in this case we shall have z D x 2 R and m Ä 2, n Ä 2, p Ä 3, q Ä 3. The contour L in (F.4.1) can be chosen as follows:

(i) L D Li1 chosen to go from i1 to Ci1 leaving to the right all the poles of P.A/, namely the poles sj;k D .bj Ck/=ˇj ; j D 1;2;:::;m; k D 0; 1; : : : of the functions appearing in A.s/, and to left all the poles of P.B/, namely the poles sj;l D .aj 1l/=ˇj ; j D 1;2;:::;n; l D 0; 1; : : : of the functions appearing in B.s/. (ii) L D LC1 is a loop beginning and ending at C1 and encircling once in the negative direction all the poles of P.A/, but none of the poles of P.B/. (iii) L D L1 is a loop beginning and ending at 1 and encircling once in the positive direction all the poles of P.B/, but none of the poles of P.A/. Braaksma has shown that, independently of the choice of L, the Mellin–Barnes integral makes sense and deﬁnes an analytic function of z in the following two cases 372 F Higher Transcendental Functions

Xq Xp >0;0

Via the following useful and important formula for the H-function Ä ˇ Ä ˇ ˇ ˇ m;n ˇ.aj ;˛j /1;p n;m 1 ˇ .1 bj ;ˇj /1;q Hp;q z ˇ D Hq;p ˇ (F.4.8) .bj ;ˇj /1;q z .1 aj ;˛j /1;p we can transform the H-function with <0and argument z to one with >0and argument 1=z. This property is useful when comparing the results of the theory of H-functions based on (F.4.1)usingzs with the theory that uses zs, which is often found in the literature. More detailed information on the existence of the H-function is presented, e.g., in [KilSai04](seealso[PrBrMa-V3]). In order to formulate this result we introduce a set of auxiliary parameters. Let m; n; p; q; ˛j ;aj ;ˇk ;bk (j D 1;:::;p; k D 1;:::;q). Deﬁne 8 ˆ Pn Pp Pm Pq ˆ a D ˛ ˛ C ˇ ˇ I ˆ j j k k ˆ j D1 j DnC1 kD1 kDmC1 ˆ Pm Pp ˆ ˆ a1 D ˇk ˛j I ˆ kD1 j DnC1 ˆ Pn Pq ˆ ˆ a D ˛j ˇkI ˆ 2 ˆ j D1 kDmC1 <ˆ Pq Pp D ˇk ˛j I ˆ kD1 j D1 (F.4.9) ˆ Qp Qq ˆ ˛j ˇk ˆ ı D ˛j ˇk I ˆ j D1 kD1 ˆ Pq Pp ˆ 1 ˆ D bk aj C .p q/I ˆ 2 ˆ kD1 j D1 ˆ Pm Pq Pn Pp ˆ D b b C a a I ˆ k k j j :ˆ kD1 kDmC1 j D1 j DnC1 1 c D m C n 2 .p C q/:

Theorem F.2 ([KilSai04, pp. 4–5]). Let the parameters a;;ı; be deﬁned as m;n in (F. 4 . 9 ). Then the Fox H-function Hp;q .z/ (as deﬁned in (F. 4 . 1 )–(F. 4 . 4 )) exists in the following cases:

L L m;n If D 1;>0; then Hp;q .z/ exists for all z W z 6D 0: (F.4.10) F.4 Fox H -Functions 373

L L m;n If D 1;D 0; then Hp;q .z/ exists for all z W 0ı: (F.4.14) L L m;n If D C1;D 0; Re <1; then Hp;q .z/ exists for z WjzjDı: (F.4.15) a If L D L ;a >0; then H m;n.z/ exists for all z Wjarg zj < ; z 6D 0: i1 p;q 2 (F.4.16) L L m;n If D i1;a D 0; C Re <1; then Hp;q .z/ exists for all z W arg z D 0; z 6D 0: (F.4.17)

Other important properties of the Fox H-functions, that can easily be derived from their deﬁnition, are included in the list below. (i) The H-function is symmetric in the set of pairs .a1;˛1/;:::;.an;˛n/, .anC1;˛nC1/;:::;.ap;˛p / and .b1;ˇ1/;:::;.bm;ˇm/, .bmC1;ˇmC1/;:::;.bq;ˇq /. (ii) If one of the .aj ;˛j /; j D 1;:::;n, is equal to one of the .bk;ˇk/; k D m C 1;:::;q; [or one of the pairs .aj ;˛j /; j D n C 1;:::;p, is equal to one of the .bk;ˇk/; k D 1;:::;m], then the H-function reduces to one of the lower order, that is, p; q and n [or m] decrease by unity. Provided n 1 and q>m,wehave Ä ˇ Ä ˇ ˇ ˇ m;n ˇ .aj ;˛j /1;p m;n1 ˇ .aj ;˛j /2;p Hp;q z ˇ D Hp1;q1 z ˇ ; .bk;ˇk /1;q1 .a1;˛1/ .bk;ˇk/1;q1 (F.4.18) Ä ˇ Ä ˇ ˇ ˇ m;n ˇ.aj ;˛j /1;p1 .b1;ˇ1/ m1;n ˇ.aj ;˛j /1;p1 Hp;q z ˇ D Hp1;q1 z ˇ : .b1;ˇ1/.bk;ˇk/2;q .bk;ˇk/2;q (F.4.19)

(iii) Ä ˇ Ä ˇ ˇ ˇ m;n ˇ.aj ;˛j /1;p m;n ˇ.aj C ˛j ;˛j /1;p z Hp;q z ˇ D Hp;q z ˇ : (F.4.20) .bk;ˇk/1;q .bk C ˇk ;ˇk/1;q

(iv) Ä ˇ Ä ˇ ˇ ˇ 1 m;n ˇ.aj ;˛j /1;p m;n c ˇ.aj ;c˛j /1;p Hp;q z ˇ D Hp;q z ˇ ;c>0: (F.4.21) c .bk;ˇk/1;q .bk;cˇk/1;q 374 F Higher Transcendental Functions

The convergent and asymptotic expansions (for z ! 0 or z !1) are mostly obtained by applying the residue theorem in the poles (assumed to be simple) of the Gamma functions appearing in A.s/ or B.s/ that are found inside the specially chosen path. In some cases (in particular if n D 0 , B.s/ D 1)wefindan exponential asymptotic behaviour. In the presence of a multiple pole s0 of order N the treatment becomes more cumbersome because we need to expand in a power series at the pole the product s N 1 of the involved functions, including z , and take the first N terms up to .s s0/ N 1 inclusive. Then the coefficient of .s s0/ is the required residue. Let us consider the case N D 2 (double pole) of interest for the fractional diffusion. Then the expansions for the Gamma functions are of the form 2 .s/D .s0/ 1 C .s0/.s s0/ C O .s s0/ ;s! s0 ;s0 ¤ 0; 1; 2;::: .1/k .s/D 1 C .k C 1/.s C k/ C O .s C k/2 ;s!k; .kC 1/.s C k/ where k D 0; 1; 2; : : : and .z/ denotes the logarithmic derivative of the function,

d 0.z/ .z/ D log .z/ D ; dz .z/ whereas the expansion of zs yields the logarithmic term

s s0 2 z D z 1 C log z .s s0/ C O..s s0/ / ;s! s0 :

F.4.2 Series Representations and Asymptotics: Recurrence Relations

Series representations of the Fox H-functions can be found by applying the Residue Theory to calculate the corresponding Mellin–Barnes integral. These calculations critically depend on the choice of the contour of integration L in (F.4.1)andthe distribution of poles of the integrand there. For simplicity, let us suppose that the poles of the Gamma functions .bk Cˇks/ and the poles of the Gamma functions .1 aj ˛j s/ do not coincide, i.e.

˛j .bk C r/ 6D ˇk.ai t 1/; j D 1;:::;n; k D 1;:::;m; r;t D 0;1;2;:::: (F.4.22) F.4 Fox H -Functions 375

This assumption allow us to choose one of the above described contours of integration L D L1,orL D LC1,orL D Li1. The series representations of Fox H-functions are based on the following theorem: Theorem F.3 ([KilSai04,p.5]). Let the conditions (F.4.22) be satisﬁed. Then the following assertions hold true. (i) In the cases (F.4.10) and (F.4.11)theH-function (F. 4 . 1 ) is analytic in z and can be calculated via the formula

Xm X1 h i m;n Hm;n s Hp;q .z/ D RessDbkr p;q .s/z ; (F.4.23) kD1 rD0

where the sum is calculated with respect to all poles bkr of the Gamma functions .bk C ˇks/. (ii) In the cases (F.4.13) and (F.4.14)theH-function (F. 4 . 1 ) is analytic in z and can be calculated via the formula

Xm X1 h i m;n Hm;n s Hp;q .z/ D RessDajt p;q .s/z ; (F.4.24) kD1 rD0

where the sum is calculated with respect to all poles ajt of the Gamma functions .1 aj ˛j s/. (iii) In the case (F.4.16)theH-function (F. 4 . 1 ) is analytic in z in the sector jarg zj < a 2 .

Theorem F.4 ([KilSai04,p.6]). Suppose the poles of the Gamma functions .bkC ˇks/ and the poles of the Gamma functions .1 aj ˛j s/ do not coincide.

(i) If all the poles of the Gamma functions .bk C ˇks/ are simple, and either m;n >0;z 6D 0,or D 0; 0 < jzj <ı, then the Fox H-function Hp;q .z/ has the power series expansion

Xm X1 m;n .bk Cr/=ˇk Hp;q .z/ D hkrz ; (F.4.25) kD1 rD0

where h i Hm;n hkr D lim .s bkr/ p;q .z/ s!bkr Qm Á Qn Á ˇi ˛i bi Œbk C r 1 ai C Œbk C r r ˇk ˇk .1/ iD1;i6Dk iD1 D Qp Á Qq Á: rŠˇk ˛i ˇi ai Œbk C r 1 bi C Œbk C r ˇk ˇk iDnC1 iDmC1 (F.4.26) 376 F Higher Transcendental Functions

(ii) If all the poles of the Gamma functions .1 aj ˛j s/ are simple, and either m;n <0;z 6D 0,or D 0; jzj >ı, then the Fox H-function Hp;q .z/ has the power series expansion

Xm X1 m;n .aj 1t/=˛j Hp;q .z/ D hjtz ; (F.4.27) kD1 rD0

where h i Hm;n hjt D lim .s ajt/ p;q .z/ s!ajt Qm Á Qn Á ˇi ˛i bi C Œ1 aj C t 1 ai C Œ1 aj C t t ˛j ˛j .1/ iD1 iD1;i6Dj D Qp Á Qq Á: tŠ˛j ˛i ˇi ai C Œ1 aj C t 1 bi Œ1 aj C t ˛j ˛j iDnC1 iDmC1 (F.4.28)

Theorem F.5 ([KilSai04,p.9]). Suppose the poles of the Gamma functions .bkC ˇks/ and the poles of the Gamma functions .1 aj ˛j s/ do not coincide. Let either <0;z 6D 0 or D 0; jzj >ı. m;n Then the Fox H-function Hp;q .z/ has the following power-logarithmic series expansion

Njt1 X0 X00 X m;n .aj 1t/=˛j .aj 1t/=˛j l Hp;q .z/ D hjtz C Hjtlz Œlog z ; (F.4.29) j;t j;t lD0 P 0 where summation in is performed over all j; t; j D 1;:::;n;t D 0; 1;P 2; : : :, 00 for which the poles of .1 aj ˛j s/ are simple, and summation in is performed over all values of parameters j; t for which the poles of .1 aj ˛j s/ have order Njt, the constants hjt are given by the formulas (F.4.28), and the constants Hjtl can be explicitly calculated (see [KilSai04,p.8]). From Theorems F.3 to F.5 follow corresponding asymptotic power- and power- m;n logarithmic type expansions at inﬁnity of the Fox H-functions Hp;q .z/ (see, e.g., [KilSai04]). Exponential asymptotic expansions in the case >0;a D 0 and in the case n D 0 are presented, for example, in [KilSai04, Sect. 1.6, 1.7] (see also [Bra62, MaSaHa10, SrGuGo82]). More detailed information on the asymptotics of the Fox H-functions at inﬁnity can be found in [KilSai04, Ch. 1]. The asymptotic behaviour of the Fox H-functions at zero is also discussed there. The following two three-term recurrence formulas are linear combinations of the H-function with the same values of parameters m; n; p; q in which some aj and bk are replaced by aj ˙ 1 and by bk ˙ 1, respectively. Such formulas are called contiguous relations in [SrGuGo82]. F.4 Fox H -Functions 377

Let m 1 and 1 Ä n Ä p 1, then the following recurrence relation holds: Ä ˇ ˇ m;n ˇ.aj ;˛j /j D1;:::;;p .b1˛p apˇ1 C ˇ1/Hp;q z ˇ .bj ;ˇj /j D1;:::;q Ä ˇ ˇ m;n ˇ .aj ;˛j /j D1;:::;p D ˛p Hp;q z ˇ .b1 C 1; ˇ1/; .bj ;ˇj /j D2;:::;q Ä ˇ ˇ m;n ˇ.aj ;˛j /j D1;:::;p1;.ap1;˛p/ ˇ1Hp;q z ˇ : (F.4.30) .bj ;ˇj /j D1;:::;q

Let n 1 and 1 Ä m Ä q 1, then the following recurrence relation holds: Ä ˇ ˇ m;n ˇ.aj ;˛j /j D1;:::;;p .bq˛1 a1ˇq C ˇq /Hp;q z ˇ .bj ;ˇj /j D1;:::;q Ä ˇ ˇ m;n ˇ.a1 1; ˛1/; .aj ;˛j /j D2;:::;;p D ˇpHp;q z ˇ .bj ;ˇj /j D1;:::;q Ä ˇ ˇ m;n ˇ .aj ;˛j /j D1;:::;p ˛1Hp;q z ˇ : (F.4.31) .bj ;ˇj /j D1;:::;q1;.bq C 1; ˇq/

A complete list of contiguous relations for H-functions can be found in [Bus72].

F.4.3 Special Cases

Most elementary functions can be represented as special cases of the Fox H- function. Let us present a list of such representations. Ä ˇ ˇ Á 1;0 ˇ 1 b=ˇ 1 H z ˇ D z exp z ˇ I (F.4.32) 0;1 .b; ˇ/ ˇ Ä ˇ ˇ 1;1 ˇ.1 a; 1/ a H z ˇ D .a/.1 C z/ D .a/1F0.aIz/I (F.4.33) 1;1 .0; 1/ Ä ˇ ˇ 1;0 ˇ.˛ C ˇ C 1; 1/ ˛ ˇ H z ˇ D z .1 z/ I (F.4.34) 1;1 .˛; 1/ Ä ˇ 2 ˇ 1;0 z ˇ 1 H0;2 ˇ 1 D p sin zI (F.4.35) 4 2 ;1 ; .0; 1/ Ä ˇ 2 ˇ 1;0 z ˇ 1 H0;2 ˇ 1 D p cos zI (F.4.36) 4 .0; 1/; 2 ;1 Ä ˇ 2 ˇ 1;0 z ˇ i H0;2 ˇ 1 D p sinh zI (F.4.37) 4 2 ;1 ; .0; 1/ 378 F Higher Transcendental Functions Ä ˇ 2 ˇ 1;0 z ˇ i H ˇ D p cosh zI (F.4.38) 0;2 4 .0; 1/; 1 ;1 Ä ˇ 2 ˇ 1;0 ˇ.1; 1/; .1; 1/ ˙H z ˇ D log .1 ˙ z/I (F.4.39) 2;2 .1; 1/; .0; 1/ " ˇ # ˇ ˇ 1 ;1 ; 1 ;1 H 1;2 z2 ˇ 2 2 D 2 arcsin zI (F.4.40) 2;2 ˇ 1 .0; 1/; 2 ;1 " ˇ # ˇ ˇ 1 ;1 ; .1; 1/ H 1;2 z2 ˇ 2 D 2 arctan z: (F.4.41) 2;2 ˇ 1 2 ;1 ; .0; 1/

A number of representation formulas relating some special functions to the Fox H-function is presented in [KilSai04, Sect. 2. 9]. Two of these formulas appear below. Ä ˇ ˇ 1;1 ˇ.0; 1/ H z ˇ D E˛;ˇ.z/; (F.4.42) 1;2 .0; 1/; .1 ˇ; ˛/ where E˛;ˇ is the Mittag-Lefﬂer function

X1 zk E .z/ D : ˛;ˇ .˛kC ˇ/ kD0 Ä ˇ ˇ 1 1;1 ˇ.1 ; 1/ H z ˇ D E .z/; Re >0; (F.4.43) ./ 1;2 .0; 1/; .1 ˇ; ˛/ ˛;ˇ

 where E˛;ˇ is the three-parametric (Prabhakar) Mittag-Lefﬂer function

X1 ./ zk E .z/ D k : ˛;ˇ .˛kC ˇ/ kD0 Ä ˇ ˇ 1;0 ˇ H z ˇ D J .z/; (F.4.44) 0;2 .0; 1/; .; / 

 where J is the Bessel–Maitland (or the Bessel–Wright) function

X1 .z/k J .z/ D : .C k C 1/kŠ kD0 Ä ˇ 2 ˇ 1;1 z ˇ. C 2 ;1/ H1;3 ˇ D J; .z/; (F.4.45) 4 . C 2 ;1/;.2 ; 1/; .. C 2 / ; / F.4 Fox H -Functions 379

where J; is the generalized Bessel–Maitland (or the generalized Bessel–Wright) function C2 C2k X1 .1/k z J .z/ D 2 : ; .C k C C 1/ .k C C 1/ kD0 2 3 Ä ˇ ˇ ˇ .al ;˛l /1;p ˇ 1;p ˇ .1 a1;˛1/;:::;.1 ap;˛p / 4 ˇ 5 Hp;qC1 z ˇ D pq ˇ z ; .0; 1/ ; .1 b1;ˇ1/;:::;.1 bq;ˇq/ .bl ;ˇl /1;q (F.4.46) where pq is the generalized Wright function 2 3 ˇ Q .a ;˛ / 1 p l l 1;p ˇ X .a C ˛ k/ zk 4 ˇ 5 Q lD1 l l pq ˇ z D q : j D1 .bj C ˇj k/ kŠ .bl ;ˇl /1;q kD0

F.4.4 Relations to Fractional Calculus

Following [KilSai04, Sect. 2.7] we present here two theorems describing the relationship of the Fox H-function to fractional calculus. Theorem F.6 ([KilSai04, pp. 52–53]). Let ˛ 2 C (Re ˛>0), ! 2 C, and >0. Let us assume that either a >0or a D 0 and Re <1. Then the following statements hold: (i) If Ä Re b min k C Re !>1; (F.4.47) 1ÄkÄm ˇk

for a >0or a D 0 and 0, while Ä Re b Re C 1=2 min k ; C Re !>1; (F.4.48) 1ÄkÄm ˇk 

˛ for a D 0 and <0, then the Riemann–Liouville fractional integral I0C of the Fox H-function exists and the following relation holds: Â Ä ˇ Ã ˇ ˛ ! m;n ˇ .a1;˛1/;:::;.ap ;˛p/ I0Ct Hp;q t ˇ .x/ .0; 1/ ; .b1;ˇ1/;:::;.bq;ˇq / Ä ˇ ˇ !C˛ m;nC1 ˇ .!; /;.a1;˛1/;:::;.ap;˛p/ D x HpC1;qC1 x ˇ : (F.4.49) .b1;ˇ1/;:::;.bq;ˇq/; .! ˛; / 380 F Higher Transcendental Functions

(ii) If Ä Re a 1 min j C Re ! C Re ˛<0; (F.4.50) 1Äj Än ˛k

for a >0or a D 0 and Ä 0, while Ä Re aj 1 Re C 1=2 min ; C Re ! C Re ˛<0; (F.4.51) 1ÄkÄm ˛j 

˛ for a D 0 and >0, then the Riemann–Liouville fractional integral I of the Fox H-function exists and the following relation holds: Â Ä ˇ Ã ˇ ˛ ! m;n ˇ .a1;˛1/;:::;.ap;˛p/ It Hp;q t ˇ .x/ .0; 1/ ; .b1;ˇ1/;:::;.bq ;ˇq/ Ä ˇ ˇ !C˛ mC1;n ˇ .a1;˛1/;:::;.ap;˛p/; .!; / D x HpC1;qC1 x ˇ : (F.4.52) .! ˛; /; .b1;ˇ1/;:::;.bq;ˇq /

Theorem F.7 ([KilSai04,p.55]). Let ˛ 2 C (Re ˛>0), ! 2 C, and >0.Letus assume that either a >0or a D 0 and Re <1. Then the following statements hold: (i) If either the condition in (F.4.44) is satisﬁed for a >0or a D 0 and 0, or the condition in (F.4.45) is satisﬁed for a D 0 and <0, then the ˛ Riemann–Liouville fractional derivative D0C of the Fox H-function exists and the following relation holds: Â Ä ˇ Ã ˇ ˛ ! m;n ˇ .a1;˛1/;:::;.ap;˛p / D0Ct Hp;q t ˇ .x/ .0; 1/ ; .b1;ˇ1/;:::;.bq;ˇq/ Ä ˇ ˇ !˛ m;nC1 ˇ .!; /;.a1;˛1/;:::;.ap ;˛p/ D x HpC1;qC1 x ˇ : (F.4.53) .b1;ˇ1/;:::;.bq;ˇq/; .! C ˛; /

(ii) If Ä Re a 1 min j C Re ! C 1 fRe ˛g <0; (F.4.54) 1Äj Än ˛k

for a >0or a D 0 and Ä 0, while Ä Re aj 1 Re C 1=2 min ; C Re ! C 1 fRe ˛g <0; (F.4.55) 1ÄkÄm ˛j F.4 Fox H -Functions 381

for a D 0 and >0,wherefRe ˛g denotes the fractional part of the ˛ number Re ˛, then the Riemann–Liouville fractional derivative D of the Fox H-function exists and the following relation holds: Â Ä ˇ Ã ˇ ˛ ! m;n ˇ .a1;˛1/;:::;.ap ;˛p/ Dt Hp;q t ˇ .x/ .0; 1/ ; .b1;ˇ1/;:::;.bq;ˇq / Ä ˇ ˇ !˛ mC1;n ˇ .a1;˛1/;:::;.ap ;˛p/; .!; / D x HpC1;qC1 x ˇ : (F.4.56) .! C ˛; /; .b1;ˇ1/;:::;.bq;ˇq /

F.4.5 Integral Transforms of H -Functions

Here we consider the Fox H-function where, in the deﬁnition (F.4.1)–(F.4.4), the poles of the Gamma functions .bk C ˇks/ and the poles of the Gamma functions .1 aj ˛j s/ do not coincide. The notation introduced in (F.4.9) will be useful for us in this subsection too. The ﬁrst result is related to the Mellin transform of the Fox H-function. This follows from Theorem F.2. and the Mellin inversion theorem (see, e.g., [Tit86, Sect. 1.5]). Theorem F.8 ([KilSai04,p.43]). Let a 0, and s 2 C be such that Ä Ä Re b 1 Re a min k < Re s< min j (F.4.57) 1ÄkÄm ˇk 1Äj Än ˛j when a >0, and, additionally

Re s C Re <1; when a D 0. Then the Mellin transform of the Fox H-function exists and the following relation holds: Â Ä ˇ Ã Ä ˇ ˇ.a ;˛ / .a ;˛ / ˇ M m;n ˇ j j 1;p Hm;n j j 1;p ˇ Hp;q x ˇ .s/ D p;q ˇ s ; (F.4.58) .bj ;ˇj /1;q .bj ;ˇj /1;q

Hm;n where p;q is the kernel in the Mellin–Barnes integral representation of the H-function. A number of more general formulas for the Mellin transforms of the Fox H- function are presented in [KilSai04,p.44],[MaSaHa10, pp. 39–40]. The next theorem gives the formula for the Laplace transform of the Fox H-function. 382 F Higher Transcendental Functions

Theorem F.9 ([KilSai04, p. 45]). Let either a >0,ora D 0; Re<1 be such that Ä Re b min k > 1; (F.4.59) 1ÄkÄm ˇk when a >0,ora D 0, 0, and " # Re b Re C 1 min k ; 2 > 1; (F.4.60) 1ÄkÄm ˇk when a D 0, <0. Then the Laplace transform of the Fox H-function exists and the following relation holds for all t 2 C; Re t>0: Â Ä ˇ Ã Ä ˇ ˇ.a ;˛ / 1 1 ˇ.0; 1/; .a ;˛ / L m;n ˇ j j 1;p m;nC1 ˇ j j 1;p Hp;q x ˇ .t/ D HpC1;q ˇ : (F.4.61) .bj ;ˇj /1;q t t .bj ;ˇj /1;q

A number of more general formulas for the Laplace transform of the Fox H- function are presented in [KilSai04, pp. 46–48].

F.5 Historical and Bibliographical Notes

The (classical) hypergeometric function 2F1 is commonly deﬁned by the following Gauss series representation (see, e.g., [NIST, p. 384])

X1 X1 .a/k .b/k k .c/ .aC k/ .b C k/ k 2F1.a; bI cI z/ D z D z : .c/kkŠ .a/ .b/ .cC k/kŠ kD0 kD0

The term hypergeometric series was proposed by J. Wallis in his book Arithmetica Infinitorum (1655). Hypergeometric series were studied by L. Euler, and a systematic analysis of their properties was presented in C.-F. Gauss’s 1812 paper (see the reprint in the collection of Gauss’s works [Gauss, pp. 123–162]). Studies in the nineteenth century included those of E. Kummer [Kum36], and the fundamental characterization by Bernhard Riemann of the hypergeometric function by means of the differential equation it satisfies. Riemann showed that the second-order differential equation for 2F1, examined in the complex plane, could be characterized (on the Riemann sphere) by its three regular singularities. Historically, confluent hypergeometric functions were introduced as solutions of a degenerate form of the hypergeometric differential equation. Kummer’s confluent hypergeometric function M.aI bI z/ (known also as the ˚-function, see [Bat-1]) was F.5 Historical and Bibliographical Notes 383 introduced by Kummer in 1837 ([Kum37]) as a solution to (Kummer’s) differential equation

d2w dw z C .b z/ C aw D 0: dz2 dz

The function M.a;bI z/ can be represented in the form of a series too

X1 .a/k k M.a;bI z/ D z D 1F1.aI bI z/: .b/kkŠ kD0

Another (linearly independent) solution to Kummer’s differential equation U.aI bI z/ (known also as the -function, see [Bat-1]) was found by Tricomi in 1947 ([Tri47]). The function .˛;ˇI z/, called the Wright function, was introduced by Wright in 1933 (see [Wri33]) in relation to the asymptotic theory of partitions. An extended discussion of its properties and applications is given in [GoLuMa99]. Special attention is given to the key role of the Wright function in the theory of fractional partial differential equations. The generalized hypergeometric functions introduced by Pochhammer [Poc70] and Goursat [Gou83a, Gou83b] are solutions of linear differential equations of order n with polynomial coefficients. These functions were considered later by Pincherle. Thus, Pincherle’s paper [Pin88] is based on what he called the “duality principle”, which relates linear differential equations with rational coefficients to linear difference equations with rational coefficients. Let us recall that the phrase “rational coefficients” means that the coefficients are in general rational functions (i.e. a ratio of two polynomials) of the independent variable and, in particular, polynomials (for more details see [MaiPag03]). These integrals were used by Meijer in 1946 to introduce the G-function into mathematical analysis [Mei46]. From 1956 to 1970 a lot of work was done on this function, which can be seen from the bibliography of the book by Mathai and Saxena [MatSax73]. The H-functions, introduced by Fox [Fox61] in 1961 as symmetrical Fourier kernels, can be regarded as the extreme generalization of the generalized hypergeometric functions pFq beyond the Meijer G functions (see, e.g. [Sax09]). The importance of this function is appreciated by scientists, engineers and statisticians due to its vast potential of applications in diverse fields. These functions include, among others, the functions considered by Boersma [Boe62], Mittag-Leffler [ML1, ML2, ML3, ML4], the generalized Bessel function due to Wright [Wri34], the generalization of the hypergeometric functions studied by Fox (1928), and Wright [Wri35b, Wri40c], the Krätzel function [Kra79], the generalized Mittag-Leffler function due to Dzherbashyan [Dzh60], the generalized Mittag-Leffler function due to Prabhakar [Pra71] and to Kilbas and Saigo [KilSai95a], the multi-index Mittag-Leffler function due to Kiryakova [Kir94]andLuchko[Luc99](seealso 384 F Higher Transcendental Functions

[KilSai96]), etc. Except for the functions of Boersma [Boe62], the aforementioned functions cannot be obtained as special cases of the G-function of Meijer [Mei46], hence a study of the H-function will cover a wider range than the G-function and gives general, deeper, and useful results directly applicable in various problems of a physical, biological, engineering and earth sciences nature, such as ﬂuid ﬂow, rheology, diffusion in porous media, kinematics in viscoelastic media, relaxation and diffusion processes in complex systems, propagation of seismic waves, anomalous diffusion and turbulence, etc. See, Caputo [Cap69], Glöckle and Nonnenmacher [GloNon93], Mainardi et al. [MaLuPa01], Saichev and Zaslavsky [SaiZas97], Hilfer [Hil00], Metzler and Klafter [MetKla00], Podlubny [Pod99], Schneider [Sch86]and Schneider and Wyss [SchWys89] and others. A major contribution of Fox involves a systematic investigation of the asymptotic expansion of the generalized hypergeometric function (now called Wright functions, or generalized Wright functions,orFox–Wright functions):

Qp .aj C ˛j l/ X1 l j D1 z F ..a ;˛ /;:::;.a ;˛ /I .b ;ˇ /;:::;.b ;ˇ /I z/ D ; p q 1 1 p p 1 1 q q Qq lŠ lD0 .bk C ˇkl/ kD1

Pq where z 2 C, aj ;bk 2 C, ˛j ;ˇk 2 R, j D 1;:::;pI k D 1;:::;qI ˇk kD1 Pp ˛j 1. His method is an improvement of the approach by Barnes (see, j D1 e.g., [Barn07b]) who found an asymptotic expansion of the ordinary generalized hypergeometric function pFq .z/. In 1961 Fox introduced [Fox61]theH-function in the theory of special functions, which generalized the MacRobert’s E-function (see [Mac-R38], [Sla66, p. 42]), the generalized Wright hypergeometricfunction, and the Meijer G-function. In the mentioned paper he investigated the far-most generalized Fourier (or Mellin) kernel associated with the H-function and established many properties and special cases of this kernel. Like the Meijer G-functions, the Fox H-functions turn out to be related to the Mellin–Barnes integrals and to the Mellin transforms, but in a more general way. After Fox, the H-functions were carefully investigated by Braaksma [Bra62], who provided their convergent and asymptotic expansions in the complex plane, based on their Mellin–Barnes integral representation. More recently, the H-functions, being related to the Mellin transforms (see [Mari83]), have been recognized to play a fundamental role in probability theory and statistics (see e.g. [MaSaHa10, Sch86, SaxNon04, UchZol99, Uch03]), in fractional calculus [KilSai99, KiSrTr06, Kir94], and its applications [AnhLeo01, AnhLeo03, AnLeSa03, Hil00, MatSax73], including phenomena of non-standard (anomalous) relaxation and diffusion [GorMai98, Koc90]. Several books specially devoted to H-functions and their applications have been published recently. Among them are F.6 Exercises 385 the books by Kilbas and Saigo [KilSai04] and by Mathai, Saxena and Haubold [MaSaHa10]. In [MaPaSa05] the fundamental solutions of the Cauchy problem for the space- time fractional diffusion equation are expressed in terms of proper Fox H-functions, based on their Mellin–Barnes integral representations.

F.6 Exercises

F.6.1. Prove that the following equalities hold for all z; jzj <=4([NIST, p. 386]): (i) F.aI 1=2 C aI 1=2Itan2 z/ D .cos z/2a cos .2az/; sin ..1 2a/z/ (ii) F.aI 1=2 C aI 3=2Itan2 z/ D .cos z/2a . .1 2a/sin z F.6.2. Prove that the following equalities hold for all z; jzj <=2([NIST, p. 386]): (i) F.aI aI 1=2I sin2 z/ D cos .2az/; cos ..2a 1/z/ (ii) F.aI 1 aI 1=2I sin2 z/ D ; cos z sin ..2a 1/z/ (iii) F.aI 1 aI 3=2I sin2 z/ D . .2a 1/ sin z F.6.3. Prove the following representations of the Kummer conﬂuent hypergeometric function 1F1 and the Gauss hypergeometric function 2F1 in terms of the Fox H-function ([KilSai04,p.63]): Ä ˇ ˇ 1;1 ˇ .1 a; 1/ .a/ .i/ H z ˇ D 1F1.aI cIz/I 1;2 .0; 1/; .1 c;1/ .c/ Ä ˇ ˇ 1;2 ˇ.1 a; 1/; .1 b; 1/ .a/ .b/ .ii/H z ˇ D 2F1.aI bI cIz/: 2;2 .0; 1/; .1 c;1/ .c/

F.6.4. Prove the following relations for the incomplete gamma- and Beta-functions ([Kir94, p. 334]):

Zz t ˛1 .i/ .˛; z/ WD e t dt D 1F1.˛I ˛ C 1Iz/I 0 Zz p1 q1 .ii/Bz.p; q/ WD t .1 t/ dt D 2F1.p; 1 qI p C 1I z/: 0 386 F Higher Transcendental Functions

F.6.5. Prove the following representation of the error function in terms of conﬂuent hypergeometric function ([Kir94,p.334]):

Zz 2 t 2 2z 1 z 2 erf.z/ WD p e dt D p 1F1. I Iz /: 2 2 0

F.6.6. Prove the following representation of the classical polynomials in terms of special cases of the hypergeometric functions pFq : (i) Laguerre polynomials ([MaSaHa10, p. 29])

ez dn ˚ « .1 C ˛/ L.˛/.z/ WD ezznC˛ D n F .nI ˛ C 1I z/I n z˛nŠ dzn nŠ 1 1

(ii) Jacobi polynomials ([MaSaHa10,p.28])

.1/n dn ˚ « P .˛;ˇ/.z/ WD .1 z/˛.1 C z/ˇ .z2 1/n n 2nnŠ dzn Â Ã .˛ C 1/ 1 z D n F n; n C ˛ C ˇ C 1I ˇ C 1I I nŠ 2 1 2

(iii) Legendre polynomials ([Kir94, p. 333]) Â 1 dn ˚ « P .z/ WD .1 z/˛Cn.1 C z/ˇCn D .1/n F n; n n 2nnŠ dzn 2 1 Ã 1 z C1I 1I ; jzj <1I 2

(iv) Tchebyshev polynomials ([Kir94, p. 333])

p Â Ã nŠ . 1 ; 1 / 1 1 z T .z/ WD cos.arccos z/ D P 2 2 .z/ D F n; nI I I n 1 n 2 1 2 2 .nC 2 /

(v) Bessel polynomials ([Kir94, p. 333])

n Á X .n/ .a C n 1/ z k Y .z;a;b/ WD k k D F n; a n kŠ b 2 0 kD0 Á z Cn 1II I b F.6 Exercises 387

(vi) Hermite polynomials ([NIST, p. 443]) Â Ã X .1/k.2z/n2k n n 1 1 H .z/ WD nŠ Œn=2 D .2z/n F ; C II : n kŠ.n 2k/Š 2 0 2 2 2 z2 kD0

F.6.7. Prove the following representations of some elementary functions in terms of the Meijer G-function ([Kir94, pp. 331–332]): Ä ˇ ˇ z 1;1 ˛ ˇ .i/ D a ˛ G az ˇ ˛ I 1 C az˛ 1;1 Ä ˇ ˛ ˇ ˇ z˛ ˇ 1;0 ˛ ˇˇ .ii/ z e D ˛ G z ˇ I 0;1 ˛ Â Ã Ä ˇ 1 C z ˇ 1 1;2 2 ˇ 1; 2 .iii/ log D G1;1 z ˇ 1 ; jzj <1I 1 z 2 ;0 Ä Ä ˇ p 12a ˇ 3 1 2a 1 1;2 ˇ 1 a; a .iv/ .1 C 1 z/ D p G z ˇ 2 : 2 222a 2;2 0; 1 2a

F.6.8. Prove the following series representation for the Fox H-function ([Bra62, p. 279]): Ä ˇ ˇ b1 1;n ˇ1 ˇ .a1;˛1/;:::;.ap ;˛p/ ˇ1z Hp;q z ˇ .b1;ˇ1/;:::;.bq;ˇq / Qn ÁÁ b1C 1 aj C ˛j X1 ˇ1 .z/ j D1 D ÁÁ ÁÁ: Š Qq Qp b1C b1C 1 bk C ˇk aj ˛j ˇ1 ˇ1 kD2 j DnC1 References

[Abe23] Abel, N.H.: Oplösning af et Par Opgaver ved Hjelp af bestemie Integraler. Magazin for Naturvidenskaberne Aargang 1, Bind 2, 11–27 (1823), in Norwe- gian. French translation “Solution de quelques problèmes à l’aide d’intégrales déﬁnies”. In: Sylov, L., Lie, S. (eds.) Oeuvres Complètes de Niels Henrik Abel (Deuxième Edition), I, Christiania, 11–27 (1881). Reprinted by Éditions Jacques Gabay, Sceaux (1992) [Abe26] Abel, N.H.: Auﬂoesung einer mechanischen Aufgabe. Journal für die reine und angewandte Mathematik (Crelle), 1, 153–157 (1826, in German). French translation “Résolution d’un problème de mécanique” In: Sylov, L., Lie, S. (eds.) Oeuvres Complètes de Niels Henrik Abel (Deuxième Edition), I, Christiania (1881), pp. 97–101. Reprinted by Éditions Jacques Gabay, Sceaux (1992). English translation “Solution of a mechanical problem” with comments by J.D. Tamarkin In: Smith, D.E. (ed.) A Source Book in Mathematics, pp. 656–662. Dover, New York (1959) [Abe26a] Abel, N.H.: Untersuchungen über die Reihe:

m m .m 1/ m .m 1/ .m 2/ 1 C x C x2 C x3 C 1 1 2 1 2 3 u. s. w. J. Reine Angew. Math. 1, 311–339 (1826); translation into English in: Researches on the series

m m.m 1/ m.m 1/.m 2/ 1 C x C x2 C x3 C : 1 1:2 1:2:3 Tokio Math. Ges. IV, 52–86 (1891) [AblFok97] Ablowitz, M.J., Fokas, A.S.: Complex Variables. Cambridge University Press, Cambridge (1997) [AbrSte72] Abramovitz, M., Stegun, I.A.: Handbook of Mathematical Functions, 10th printing. National Bureau of Standards, Washington, DC (1972) [Adv-07] Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineer- ing. Springer, Berlin (2007) [Aga53] Agarwal, R.P.: A propos d’une note de M. Pierre Humbert. C. R. Acad. Sci. Paris 236, 2031–2032 (1953)

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Abel equation of the second kind, 168 Contiguous relations, 376 Abel integral equation of the first kind, 165 Continuous time random walk (CTRW), 239 Abel–Riemann fractional derivative, 334 Correspondence principle, 222 Abel–Riemann fractional integral, 334 Counting number process, 238 Acta Mathematica,8 Creep compliance, 220 ˛-exponential function, 83 Angular density, 289 Angular symmetry, 289 Diffusion limit, 254 Asymptotic universality of Mittag-Leffler Dirichlet condition, 301 waiting time density, 263 Discrete Mittag-Leffler distribution, 264

Basset problem, 214 Empty product convention, 106 Bernstein function, 237 Entire function, 285 Bessel polynomials, 386 Entire function of completely regular growth, Bessel–Wright function, 363 289 Beta function, 278 Erlang process, 239 Bromwich inversion formula, 41 Error function, 20 Euler integral of the second kind, 278 Euler–Mascheroni constant, 278 Caputo derivative, 336 Euler transform of the Mittag-Leffler function, Caputo–Dzherbashyan derivative, 337 36 Cauchy problem for OFDEs, 174 Extended generalized Mittag-Leffler function, Cauchy type initial conditions, 172 134 Cauchy type problem for OFDEs, 172 Classical Kolmogorov–Feller equation, 242 Cole–Cole response function, 226 Failure probability, 240 Completely monotonic function, 46 Four-parametric Mittag-Leffler function, 129 Complex factorial function, 270 Fourier convolution, 302 Composite fractional oscillation equation, 212 Fourier integral formula, 301 Composite fractional relaxation and oscillation Fourier integral theorem, 301 equations, 211 Fourier transformation, 300 Composite fractional relaxation equation, Fourier transform operator, 300 212 Fox H -functions, 14, 370 Compound renewal processes, 265 Fractional anti-Zener model, 223 Constitutive equations, 220 Fractional equations for heat transfer, 227

Fractional KdV equation, 232 Legendre’s duplication formula, 273 Fractional Klein–Gordon equation, 231 Liouville fractional integral, 330 Fractional Maxwell model, 222 Liouville–Weyl fractional derivatives, 331 Fractional Newton equation, 225 Fractional Newton (Scott Blair) model, 222 Fractional Ohm law, 225 Material functions, 220 Fractional operator equation, 223 Maxwell type fractional equation, 232 Fractional oscillation, 208 Meijer G-functions, 14, 365 Fractional Poisson process, 247 Mellin–Barnes integral, 319 Fractional relaxation, 208 Mellin–Barnes integral formula, 13 Fractional Voigt model, 222 Mellin convolution formula, 308 Fractional Zener model, 222 Mellin fractional analogue of Green’s function, 181 Mellin integral transform, 307 Gamma function, 269 Miller–Ross function, 58 Generalized Bessel–Wright function, 363 Mittag-Leffler distribution, 264 Generalized hypergeometric functions, 360 Mittag-Leffler function, 11 Generalized Kolmogorov–Feller equation, 242 Mittag-Leffler functions with 2n parameters, Grünwald–Letnikov fractional derivative, 346 145 Mittag-Leffler, Gösta Magnus, 7 Mittag-Leffler integral representation, 25 Hankel integral representation, 275 Multiplication formula for -function, 273 Hankel integration path, 12 Hankel loop, 74 Hankel path, 274 Non-local operators, 329 Heaviside function, 220 Hermite polynomials, 359, 387 Operational calculus for FDEs, 174 Hypergeometric differential equation, 355 Order of an entire function, 286 Hypergeometric function, 351 Ordinary fractional differential equation, 172

Incomplete gamma function, 21 Parseval identity, 302 Indicator function of an entire function, 287 Phragmén–Lindelöf theorem, 13 Inﬁnite product, 287 Pochhammer’s symbol, 274 Inverse Fourier operator, 301 Pot, 221 Inverse Mellin transform, 307 Prabhakar function, 97 Progressive/regressive fractional derivative, 329 Jacobi polynomials, 386

Quasi-polynomial, 177 Kilbas–Saigo function, 106 Kummer conﬂuent hypergeometric function, 357 Rabotnov function, 58 Relaxation differential equation, 202 Relaxation modulus, 220 Laguerre polynomials, 360, 386 Renewal function, 239 Laplace–Abel integral, 10 Renewal process, 238 Laplace abscissa of convergence, 40, 304 Rescaling concept, 262 Laplace convolution, 306 Respeeding concept, 262 Laplace image, 40, 304 Riemann–Liouville fractional integral, 327 Laplace original, 40, 304 Riesz fractional derivative, 339 Laplace transform method for FDEs, 177 Riesz fractional integral, 339 Legendre polynomials, 386 Riesz potential, 339 Index 443

Semigroup property, 328 Transcendental entire functions, 285 Simple fractional oscillation equation, 202 Tricomi conﬂuent hypergeometric function, Simple fractional relaxation equation, 202 357 Stirling’s formula, 277 Triplication formula, 274 Stress–strain relation, 220 Two-parametric Mittag-Lefﬂer function, 15 Survival probability, 240 Type of an entire function, 286

Tchebyshev polynomials, 386 Thinning, 251 Weierstrass elementary factors, 287 Three-parametric Mittag-Lefﬂer function, 15 Weyl fractional integral, 330 Time fractional Kolmogorov–Feller equation, Whittaker function, 102 245 Wright function, 361