arXiv:1405.0507v1 [math.CA] 29 Apr 2014 M ahmtc ujc lsicto (2000) Classification Subject Mathematics AMS calculus. neutrix using Abstract. ico a rtt pl h oyam ucin npyis h su the applicatio physics, some containing in are series integrals of functions of summation the evaluation polygamma Recently, the relevant. the and series apply rational to of first was Kirchoff integrals. hyp logarithmic-trigonometric polylogarithms, e and functions, the tions, zeta gamma, in the the are functions, arises appear polygamma functions typically and which special functions Special and elementary sums. Euler of combination rich A limit. Neutrix Neutrix, constant, Euler Functions, Polygamma phrases and words Key ¨ li rsne w omlefrthe for formulae two K¨olbig presented [2]. calculations hyas ple eti aclst unu edter,otiigfi 12] pertur [11, obtaining theory, see the calculations, field in loop quantum cofficients the to the in w calculus for malization conjuction neutrix results applied in quantum finite also to calculus, obtain [3], They neutrix see to the Corput particular, applied der van in Dam by van developed H. integral, f Hadamard and special Ng and Jack distributions Y. 21] t of 20, gave context 8, Fisher in 7, asym exploited 5, limit, from [4, been neutrix quantities has infinite the unwanted and and of sions discarding neutrix the [14]. the diverge for see of principle part, the concepts finite from Hadamard the extracted the Using value as to finite referred resulting usually quantities the infinite and defined Hadamard appropriately neglecting of technique The [2]. in functions polygamma the over oyoaih ucin e 1,1] oe osdrdtesm ove sums the ( considered with Coffey summand containing 17]. function, [16, see function, polylogarithm plctoso eti aclst pca ucin in Functions Special to Calculus Neutrix of Applications nti ae edfieteplgmafunctions polygamma the define we paper this In ojnto ihPlgmaFunctions Polygamma with Conjunction am ucin eaFnto,DgmaFunction, Digamma Function, Beta Function, Gamma : .Introduction 1. Emin ± 1) n ψ 1 ψ Ozc ¨ ( n ( ) n ( . p/q + a˘g p/q yuigtesre ento of definition series the using by ) ) /n ψ 2 ( 31,33B20, 33B15, : n aeetnint sums to extension made and ψ n ) ( ( n z ) ( a rsni Feynman in arisen was ) x o eaieitgr by integers negative for ) remti func- ergeometric sta r still are that ns h digamma the r a eie by devised was ea digamma beta, ttcexpan- ptotic edtheories, field ainseries. bation titga is integral nt ntos see unctions, auto of valuation ierenor- nite egeneral he mmation t the ith In the following we let N be the neutrix [3] having domain N ′ = {ε : 0 <ε< ∞} and range N ′′ the real numbers, with negligible functions finite linear sums of the functions − ελ lnr 1 ε, lnr ε (λ< 0, r =1, 2,...) (1) and all functions f(ε) which converge to zero in the normal sense as ε tends to zero. If f(ε) is a real (or complex) valued function defined on N ′ and if it is possible to find a constant c such that f(ε) − c is in N, then c is called the neutrix limit of f(ε) as ε → 0 and we write N−limε→0 f(ε)= c. Note that if a function f(ε) tends to c in the normal sense as ε tends to zero, it converges to c in the neutrix sense. Also note that if a function H(ε)= υ(ε)+ f(ε), where υ(ε) is the sum of negligible functions of H(ε), then p.f. H(ε), Hadamard’s finite part of H(ε), is equal to f(ε) and so N−lim H(ε) = lim f(ε) = lim p.f.H(ε). ε→0 ε→0 ε→0 The reader may find the general definition of the neutrix limit with some examples in [3, 4, 5]. In this paper we use Fisher’s principle to define the derivative of the digamma function for negative integers. First of all, we give the definition of the gamma function for all x. The gamma function Γ(x) is usually defined for x> 0 by

∞ Γ(x)= tx−1e−t dt (2) Z0 the integral only converging for x> 0, see [10, 22]. It follows from equation (2) that

Γ(x +1)= xΓ(x) (3) for x> 0 and this equation is used to define Γ(x) for negative, non-integer values of x. Using the regularization, Gelfand and Shilov [10] define the gamma function

1 n−1 ti ∞ n−1 (−1)i Γ(x)= tx−1 e−t − (−1)i dt + tx−1e−t dt + 0 i! 1 i!(x + i) Z h Xi=0 i Z Xi=0 for x> −n, x =06 , −1, −2,..., −n + 1 and

∞ n−1 ti Γ(x)= tx−1 e−t − (−1)i dt 0 i! Z h Xi=0 i for −n

Fisher proved that ∞ Γ(x) = N−lim tx−1 e−t dt → ε 0 Zε

2 x =06 , −1, −2,... and defined Γ(−m) by

∞ Γ(−m) = N−lim t−m−1 e−t dt → ε 0 Zε ∞ 1 m (−t)i m−1 (−1)i = t−m−1e−t dt + t−m−1 e−t − dt − (4) 1 0 i! i!(m − i) Z Z h Xi=0 i Xi=0 for m =1, 2,..., see [6]. More generally, the r-th derivative of gamma function Γ(x) is defined by

∞ Γ(r)(x) = N−lim tx−1 lnr t e−t dt (5) → ε 0 Zε for all x and r =0, 1, 2,..., see [8]. Fisher obtained from equation (4) that

1 (−1)m Γ(−m)+ Γ(−m +1)= m mm! from which it followed by induction that

(−1)m Γ(−m)= [φ(m) − γ] (6) m! for m =1, 2,..., where γ denotes Euler’s constant and

0, m =0, φ(m)= m 1 ( i=1 i , m =1, 2,... P in particular Γ(0) = Γ′(1) = −γ, see [6].

′ d Γ (x) The digamma function ψ(x)= dx log Γ(x)= Γ(x) defined for x> 0 has the integral representation ∞ e−t − e−xt ψ(x)= −γ + −t dt Z0 1 − e and it can be written as 1 1 − tx−1 ψ(x)= −γ + dt (x> 0). (7) Z0 1 − t

Differentiating equation (3), we have

Γ′(x +1)= Γ(x)+ xΓ′(x) (8) for x =06 , −1, −2,... and it follows that 1 ψ(x +1)= ψ(x)+ x

3 see [18, 19] and this equation is used to define ψ(x) for negative, noninteger values of x. Thus if −m

1 1 − tx−1+m m−1 1 ψ(x)= −γ + dt − (9) 0 1 − t x + k Z kX=1

In [9] Fisher and kuribayashi proved the following equations to define ψ(−m) for m =1, 2,.... Theorem 1.1.

Γ(n)(x) = N−lim Γ(n)(x + ε) ε→0 Γ(x +1) = N−lim(x + ε)Γ(x + ε) ε→0 Γ′(x +1) = N−lim[Γ(x + ε)+(x + ε)Γ′(x + ε)] ε→0 for all x and n =0, 1, 2,.... Theorem 1.1 suggested that the digamma function ψ(−m) can be defined by

Γ′(−m + ε) ψ(−m) = N−lim ε→0 Γ(−m + ε) for m =0, 1, 2,..., provided the neutrix limit exists, and with this definition

ψ(−m)= ψ(1) + φ(m)= −γ + φ(m) (10) for m =0, 1, 2,..., see [9]. Recently Tuneska and Jolevski used the integral representation of the digamma function to obtain same result given in [13].

2. Defining Polygamma Function ψ(n)(−m)

The polygamma function is defined by

dn dn+1 ψ(n)(x)= ψ(x)= ln Γ(x) (x> 0). (11) dxn dxn+1 It may be represented as

∞ n −xt (n) n+1 t e ψ (x)=(−1) −t Z0 1 − e which holds for x> 0, and

1 tx−1 lnn t ψ(n)(x)= − dt. (12) Z0 1 − t

4 It satisfies the recurrence relation (−1)nn! ψ(n)(x +1)= ψ(n)(x)+ (13) xn+1 see [1, 16, 17, 18, 22]. This is used to define the polygamma function for negative non-integer values of x. Thus if −m

1 tx+m−1 m−1 (−1)nn! ψ(n)(x)= − lnn t dt − . (14) 0 1 − t (x + k)n+1 Z kX=0

1 λ−1 −ν m K¨olbig gave the formulae for the integral 0 t (1−t) ln t dt for integer and half- integer values of λ and ν in [15]. As theR integral representation of the polygamma function is similar to the integral mentioned above, by using the neutrix limit we prove the existence of the integral in (12) as follows. 1 Now we let N be a neutrix having domain the open interval {ǫ : 0 <ε< 2 } with the same negligible functions as in equation (1). We first of all need the following lemma. Lemma 2.1 The neutrix limits as ε tends to zero of the functions

1/2 1−ε tx lnn t lnr(1 − t) dt, (1 − t)x lnn t lnr(1 − t) dt Zε Z1/2 exists for n, r =0, 1, 2,... and all x. Proof. Suppose first of all that n = r =0. Then

−x−1 x+1 1/2 2 −ε , x =6 −1, tx dt = x+1 Zε ( − ln 2 − ln ε, x = −1

1/2 x and so N−lim ε t dt exists for all x. ε→0 R 1/2 x n Now suppose that r = 0 and that N−lim ε t ln t dt exists for some nonnegative ε→0 integer n and all x. Then R

−x− n x n −2 1 ln +1 2−ε +1 ln +1 ε n+1 1/2 x n 1/2 x+1 − x+1 ε t ln t dt, x =6 −1, x n+1 t ln t dt =  R ε  − n n+2 − n+2 Z  ( 1) ln 2 ln ε m+2 , x = −1   1/2 x n and it follows by induction that N−lim ǫ t ln t dt exists for n = 0, 1, 2,... and ε→0 all x. R Finally we note that we can write

∞ r i ln (1 − t)= αint Xi=1

5 for r =1, 2,..., the expansion being valid for |t| < 1. Choosing a positive integer k such that x + k > −1, we have

1/2 tx lnn t lnr(1 − t) dt = Zε k−1 1/2 ∞ 1/2 x+i n x+i n = αin t ln t dt + αin t ln t dt. ε ε Xi=1 Z Xi=k Z It follows from what we have just proved that

k−1 1/2 x+i n N−lim αin t ln t dt ε→0 ε Xi=1 Z exists and further

∞ 1/2 ∞ 1/2 x+i n x+i n N−lim αin t ln t dt = lim αin t ln t dt ε→0 ε ε→0 ε Xi=k Z Xi=k Z ∞ 1/2 x+i n = αin t ln t dt, 0 Xi=k Z 1/2 x n r proving that the neutrix limit of ε t ln t ln (1 − t) dt exists for n, r = 0, 1, 2,... and all x. Making the substitutionR 1 − t = u in

1−ε (1 − t)x lnn t lnr t dt, Z1/2

1−ε x n r it follows that 1/2 (1 − t) ln t ln t dt also exits for n, r =0, 1, 2,... and all x. R 1 −m−1 n Lemma 2.2 The neutrix limit as ε → 0 of the integral ε t ln t dt exists for m, n =1, 2,... and R 1 n! N−lim t−m−1 lnn t dt = − . (15) → n+1 ε 0 Zε m Proof. Integrating by parts, we have

1 1 t−m−1 ln t dt = m−1ε−m ln ε + m−1 t−m−1 dt Zε Zε and so 1 1 N−lim t−m−1 ln t dt = − → 2 ε 0 Zε n proving equation (15) for n = 1 and m = 1, 2,.... Now assume that equation (15) holds for some m and n =1, 2,.... Then

1 n 1 − t−m−2 lnn t dt = (m + 1)−1ε−m−1 lnn ε + t−m−2 lnn 1 t dt Zε m +1 Zε n −(n − 1)! = (m + 1)−1ε−m−1 lnn ε + m +1 (m + 1)n

6 and it follows that 1 n! N−lim t−m−2 lnn t dt = − → n+1 ε 0 Zε (m + 1) proving equation (15) for m + 1 and n =1, 2,.... Using the regularization and the neutrix limit, we prove the following theorem. Theorem 2.3 The function ψ(n)(x) exists for n =0, 1, 2,..., and all x. Proof. Choose positive integer r such that x> −r. Then we can write

1−ε tx−1 1/2 1 r−1 lnn t dt = tx−1 lnn t − (−1)iti dt ε 1 − t ε 1 − t Z Z h Xi=0 i r−1 1/2 1−ε tx−1 + (−1)i tx+i−1 lnn t dt + lnn t dt. ε 1/2 1 − t Xi=0 Z Z We have 1/2 tx−1 1 r−1 lim lnn t − (−1)iti dt = ε→0 ε 1 − t 1 − t Z h Xi=0 i 1/2 tx−1 1 r−1 = lnn t − (−1)iti dt 0 1 − t 1 − t Z h Xi=0 i and 1−ε tx−1 1 tx−1 lim lnn t dt = lnn t dt ε→0 Z1/2 1 − t Z1/2 1 − t the integrals being convergent. Further, from the Lemma 2.1 we see that the neutrix limit of the function r−1 1/2 (−1)i tx+i−1 lnn t dt ε Xi=0 Z exists and implying that 1−ε tx−1 N−lim lnn t dt → ε 0 Zε 1 − t exists. This proves the existence of the function ψ(n)(x) for n = 0, 1, 2,..., and all x. Before giving our main theorem, we note that

1 tx−1 ψ(n)(x)= − N−lim lnn t dt → ε 0 Zε 1 − t since the integral is convergent in the neighborhood of the point t =1. Theorem 2.4 The function ψ(n)(−m) exists and

m n! ψ(n)(−m)= +(−1)n+1n!ζ(n +1) (16) in+1 Xi=1 7 for n =1, 2,... and m =0, 1, 2,..., where ζ(n) denotes zeta function. Proof. From Theorem 2.3, we have

1 t−m−1 lnn t ψ(n)(−m) = − N−lim dt → ε 0 Zε 1 − t 1 m+1 = − N−lim t−i + (1 − t)−1 lnn t dt. (17) ε→0 ε Z h Xi=1 i 1 −i n We first of all evaluate the neutrix limit of integral ε t ln t dt for i =1, 2,... and n =1, 2,.... R It follows from Lemma 2.2 that 1 n! N−lim t−i lnn t dt = − (i> 1). (18) → n+1 ε 0 Zε (i − 1) For i =1, we have 1 t−1 lnn t dt = O(ε). Zε Next 1 lnn t 1 lnn t ∞ 1 dt = dt = tk lnn t dt ε 1 − t 0 1 − t 0 Z Z kX=0 Z ∞ (−1)nn! = (k + 1)n+1 kX=0 = (−1)nn!ζ(n + 1), (19) where 1 n k n (−1) n! t ln t dt = n+1 . Z0 (k + 1) It now follows from equations (17),(18) and (19) that

1 t−m−1 lnn t ψ(n)(−m) = − N−lim dt → ε 0 Zε 1 − t m+1 1 1 lnn t = − N−lim t−i lnn t dt − N−lim dt. ε→0 ε ε→0 ε (1 − t) Xi=1 Z Z m n! = +(−1)n+1n!ζ(n + 1) in+1 Xi=1 implying equation (16).

Note that the digamma function ψ(x) can be defined by

1 1 − tx−1 ψ(x)= −γ + N−lim dt (20) → ε 0 Zε 1 − t for all x.

8 Using Lemma 2.1 we have

1 1 − t−m−1 m+1 1 dt = − t−i dt ε 1 − t ε Z Xi=1 Z m+1 [1 − ε−i+1] = −[ln 1 − ln ε] − −i +1 Xi=2 and it follows from equation (20) that

1 1 − t−m−1 m ψ(−m) = −γ + N−lim dt = −γ + i−1 ε→0 ε 1 − t Z Xi=1 = −γ + φ(m) which was obtained in [9] and [13].

References

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[5] Fisher, B., A non-commutative neutrix product of distributions, Math. Nachr., 108(1982) 117-127.

[6] Fisher, B., On defining Γ(−n) for n = 0, 1, 2,..., Rostock. Math. Kolloq., 31(1987) 4-10.

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[9] Fisher, B. and Kuribayashi, Y., Some results on the Gamma function, J. Fac. Ed. Tottori Univ. Mat. Sci., 3(2)(1988) 111-117.

9 [10] I. M. Gel’fand and G. E. Shilov, Generalized Functions, Vol. I., Academic Press, Newyork/London, (1964).

[11] Ng, Jack Y. and van Dam, H., Neutrix Calculus and Finite Quantum Field Theory, J. Phys., A; Math. Gen. 38(2005) 317-323.

[12] Ng, Jack Y. and van Dam, H., An application of neutrix calculus to Quantum Field Theory, Internat. J. Modern Phys., A21(2)(2006) 297-312.

[13] Jolevska-Tuneska, B. and Jolevski, I., Some results on the digamma function, Appl. Math. Inf. Sci., 7(2013) 167-170.

[14] Jones, D. S., Hadamard’s Finite Part, Math. Methods Appl. Sci., 19(1996) 1017-1052.

1 ν−1 −λ m [15] K¨olbig, K. S., On the integral 0 x (1 − x) ln x dx, J. Comput. Appl. Math., 18(1987) 369-394. R [16] K¨olbig, K. S., The polygamma function and the derivatives of the cotangent function for rational arguments, CERN-IT-Reports, CERN-CN-96-005, 1996.

[17] K¨olbig, K. S.,The polygamma function ψ(k)(x) for x = 1/4 and x = 3/4, J. Comput. Appl. Math., 75(1996) 43-46.

[18] Laforgia, A. and Natalini, P., Exponentials, gamma and polygamma functions: Simple proofs of classical and new inequalities, J. Math. Anal. Appl. 407(2013) 497-504.

[19] Medina, A. L. and Moll, V. H., The integrals in Gradshteyn and Ryzhik,. Part 10: The digamma function, Scientia, Series A: Math. Sci., 17(2009) 45-66.

[20] Ozc¨ .a˘g, E., Ege, I.,˙ G¨urc.ay, H. and Jolevska-Tuneska, B., Some remarks on the incomplete gamma function, in: Kenan Ta¸set al. (Eds.), Mathematical Methods in Engineering, Springer, Dordrecht, 2007, pp. 97-108.

[21] Ozc¨ .a˘g, E., Ege, I.,˙ G¨urc.ay, H. and Jolevska-Tuneska, B., On partial derivatives of the incomplete beta function, Appl. Math. Lett., 21(2008) 675-681.

[22] Rradshhteyn, I. S. and Ryzhik, I. M., Tables of integrals, Series, and Products, Academic Press, San Diego, 2000.

Emin Ozc¨ .a˘g Department of mathematics Hacettepe University Beytepe 06800 Ankara, Turkey e-mail : [email protected]

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