Some Identities with Generalized Hypergeometric Functions

Appl. Math. Inf. Sci. 10, No. 5, 1729-1734 (2016) 1729 Applied Mathematics & Information Sciences An International Journal

http://dx.doi.org/10.18576/amis/100511

Vasily E. Tarasov∗ Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991, Russia

Received: 17 Aug. 2015, Revised: 28 Feb. 2016, Accepted: 29 Feb. 2016 Published online: 1 Sep. 2016

Abstract: In this paper, we consider some identities of a special form of the generalized hypergeometric functions 1F2(a;b,c;z) of real argument z. This special form is important for application in fractional calculus and fractional dynamics. The suggested functions stand out among other generalized hypergeometric functions by the power-law form of its Fourier transforms. Identities for inﬁnite series and integrals, which include these generalized hypergeometric functions, are proved.

Keywords: Special functions, generalized hypergeometric function, fractional calculus

1 Introduction 2 Deﬁnition of generalized hypergeometric function

Generalized hypergeometric function is one of the most important special functions. Moreover the generalized In this section we give a definition and some properties of hypergeometric function includes many other functions as the generalized hypergeometric function to fix notation special cases, such as the Bessel functions, the classical for further consideration. orthogonal polynomials and elementary functions. This function has been proposed as a generalization of the Definition. The generalized hypergeometric function Gauss and Kummer hypergeometric functions [1,2]. The is define by the equation generalized hypergeometric functions have a lot of application in physics [3]. Special functions including ∞ k generalized hypergeometric functions play an important (a1)k ...(ap)k z F (a ,...,a ;b ,...,b ;z) := ∑ , (1) role in the fractional calculus [4,5,6,7] that has a long p q 1 p 1 q k=0 (b1)k ...(bq)k k! history [8]. The fractional calculus as a theory of differentiation and integrations of non-integer orders has C C a wide application in mechanics and physics (for where al ∈ (l = 1,..., p), b j ∈ and b j 6= 0,−1,−2... example, see [9] and references therein) since it allows us ( j = 1,...,q), and (a)k is the Pochhammer symbol (rising to describe processes in nonlocal and hereditary media. factorial) that is defined by Recently the generalized hypergeometric functions have been suggested to describe lattice models with power-law Γ (a + k) (a) = 1, (a) = = long-range interactions [10,11,12]. 0 k Γ (a) In this paper, we consider some properties and a(a + 1)(a + 2)...(a + k − 1) (k ∈ N). (2) identities of a special form of the function 1F2(a;b,c;z) that are important for application in fractional dynamics of media with power-law nonlocality and corresponding discrete models. Mathematically the considered functions stand out among other the generalized hypergeometric The series (1) is absolutely convergent for all values of functions by the power-law form of its Fourier transforms. z ∈ C if p ≤ q.

∗ Corresponding author e-mail: [email protected]

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−1 The function (1) has an integral representation in terms and the inverse Fourier series transform F∆ in the form of the Mellin-Barnes contour integral +π −1 1 ikn F∆ { fˆ(k)} := dke fˆ(k). (10) ∏q Γ 2π Z π 1 j 1 (b j) − F (a ,...,a ;b ,...,b ;z)= = p q 1 p 1 q π ∏p Γ 2 i l=1 (al ) Note that we use the minus sign in the exponent of (9) γ+i∞ ∏p Γ a s instead of plus that is usually used. l=1 ( l − ) −s Γ q (−z) (s)ds, (3) The Fourier series transforms of these generalized Zγ i∞ ∏ Γ (b − s) − j=1 j hypergeometric functions with n ∈ N are given by the following proposition. where |arg(−z)| < π and the path of integration separates all the poles s = −k (k ∈ N0) to the left and all the poles N Proposition 1. The Fourier series transforms of the s = a j + n (n ∈ 0, j = 1,..., p) to the right. generalized hypergeometric functions (8) have the forms In this paper, we are interested in a special form of the 1F2 function +∞ F α −ikn ∞ k ∆ {T+[ ,n]} = ∑ e (a)k z n=−∞ 1F2(a;b,c;z) := ∑ . (4) 2 2 k=0 (b)k (c)k k! α + 1 α + 3 1 π n α + 1 α 1F2 ; , ;− = |k| , (11) 2 2 2 4 πα Note the symmetry of this function

1F2(a;b,c;z)= 1F2(a;c,b;z). (5) +∞ −ikn F∆ {n T−[α,n]} = ∑ e For generalized hypergeometric function (4) the following n=−∞ differentiation relation holds α + 1 α + 3 3 π2 n2 n 1F2 ; , ;− = dn 2 2 2 4 1F2(a;b,c;x)= α + 1 dxn α−1 − i πα sgn(k)|k| , (12) (a)n 1F2(a + n;b + n,c + n;x) (n ∈ N). (6) α (b)n (x)n where > −1. It is possible to write the product formula for this function Proof. To prove (11)and (12), we use the equations π −1 α 1 α 1F2(a1;b1,c1;z) 1F2(a2;b2,c2;z)= F∆ |k| = k cos(nk)dk = π Z0 ∞ k k (a )m (a ) z ∑ ∑ 1 2 k−m . (7) πα α α π2 2 k=0 m=0 (b1)m (b2)k−m (c1)m (c2)k−m m!(k − m)! + 1 + 3 1 n = 1F2 ; , ;− , (13) α + 1 2 2 2 4 In this paper, we consider some identities of the α following generalized hypergeometric functions where > −1 and π 2 2 −1 α 1 α α + 1 α + 3 1 π n F∆ i sgn(k)|k| = − k sin(nk)dk = T [α,n] := 1F2 ; ,1 ∓ ;− , (n ∈ Z). π Z0 ± 2 2 2 4 (8) α π +1 n α + 2 α + 4 3 π2 n2 We denote this function by T [α,n] for simpliﬁcation. ± = − α 1F2 ; , ;− , (14) These functions stand out among other the generalized + 2 2 2 2 4 hypergeometric functions by the power-law form of its where α > −2. Applying the Fourier series transform Fourier transform. F∆ , which is deﬁned by (9), to equations (13) and (14), we get expression (11)and (12).

3 Fourier series transform of generalized Remark 1. The Fourier series transforms of the α hypergeometric function functions T±[ ,n] have been represented in the form

+∞ Let us consider the Fourier series transform F∆ that is −ikn F∆ T [α,n] = ∑ e T [α,n]= deﬁned by the equation + + n=−∞ ∞ +∞ −ikn 2 ∑ T [α,n] cos(kn)+ T [α,0], (15) F∆ { f [n]} := ∑ e f [n], (9) + + n=1 n=−∞

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+∞ −ikn F∆ nT−[α,n] = ∑ e nT−[α,n]= following proposition. n=−∞ ∞ Proposition 3. The generalized hypergeometric α − 2i ∑ nT−[ ,n] sin(kn). (16) functions (8) satisfy the following identities n=1 Note that +∞ ∑ k (k − n)T−[α,k]T−[β,k − n]= T±[α,0]= 1. (17) k=−∞ Proposition 1 of the Fourier transforms of the (α + 1)(β + 1) − T [α + β − 2,n], (21) generalized hypergeometric functions (8) will be used to π2 (α + β − 1) + prove identities for these functions. where α > −1, β > −1, α + β − 2 > −1.

4 Identities with generalized hypergeometric +∞ α β functions ∑ k T−[ ,k]T+[ ,k − n]= k=−∞ (α + 1)(β + 1) One of the main results of this paper are the following T [α + β,n], (22) propositions for the generalized hypergeometric functions (α + β + 1) − (8). where α > −1, β > −1, α + β > −1. Proposition 2. The following identities with the generalized hypergeometric functions hold +∞ ∑ (k − n)T+[α,k]T−[β,k − n]= ∞ +∞ α α π2 2 k=− + 1 + 3 1 k α β ∑ 1F2 ; , ;− ( + 1)( + 1) 2 2 2 4 T−[α + β,n], (23) k=−∞ (α + β + 1) β + 1 β + 3 1 π2 (k − n)2 (α + 1)(β + 1) 1F2 ; , ;− = where α 1, β 1, α β 1. 2 2 2 4 α + β + 1 > − > − + > − 2 2 α + β + 1 α + β + 3 1 π n Proof. The proof is similar to the proof of Proposition 1F2 ; , ;− , (18) 2 2 2 4 2. To prove the identities (21)-(23), we used relations (11) and (12) of the Fourier series transform of the functions where α > −1, β > −1, α + β > −1. (8). As a result, we get the identities.

Proof. Using relation (11), the Fourier series transforms of the left side and the right side of equation (18) gives 5 Simple examples of identities α β + 1 α + 1 β For α = 2, the generalized hypergeometric functions πα |k| β |k| = π T±[α,n] have the simple forms α β α β ( + 1)( + 1) + + 1 α+β |k| . (19) π2 2 n α + β + 1 πα+β 3 5 3 n 3 (−1) T [2,n]= 1F2 ; , ;− = − (24) − 2 2 2 4 π2 n n As a result we obtain the identity. and Remark 2. Using notations (8), the suggested identity π2 2 n (18) can be written in the form 3 5 1 n 3 2(−1) T [2,n]= 1F2 ; , ;− = − (25) + 2 2 2 4 π2 n2 +∞ ∑ T+[α,k]T+[β,k − n]= where n 6= 0. For n = 0, we have (17). Then, equality (21) k=−∞ can be represented by equations of the following (α + 1)(β + 1) proposition. T [α + β,n], (20) α + β + 1 + Proposition 4. For α = β = 2, identity (21) with the where α > −1, β > −1, α + β > −1. functions (24) and(24) has the form There are other identities that involving the +∞ 3 generalized hypergeometric functions (8). Using ∑ k (k − n)T [2,k]T [2,k − n]= − T [2,n] (26) α − − π2 + notations T±[ ,n], these identities are represented by the k=−∞

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that can be written as and ∞ 1 2 5 7 1 π2 n2 ∑ = 2 (n 6= 0). (27) k (k − n) n T [4,n]= 1F2 ; , ;− = k=−∞ + 2 2 2 4 k6=0,k6=n 5 4π2 (−1)n 24(−1)n For n = 0, we have − (n 6= 0), (37) π4 n2 n4 ∞ 1 π2 ∑ = . (28) 2 and T±[4,0]= 1. k=−∞ k 3 k6=0 Remark 4. Using equation 2.5.3.5 of [14] we can give the representation of the functions (8) with α = m ∈ N in the form Proof. Equality (27) can be proved directly by splitting sum on three sums T+[m,n]= ∞ ∞ 1 1 m + 1 (−1)n+2 [(m−1)/2] (−1)k m! ∑ = ∑ + ∑ (π n)m−2k−1+ k (k − n) k (k − n) πm+1 m+1 k=−∞ k=n+1 n k=0 (m − 2k − 1)! k6=0,k6=n (−1)[(m+1)/2] m! n−1 1 −∞ 1 + 2[(m + 1)/2] − m , (38) ∑ + ∑ . (29) nm+1 k (k − n) k (k − n) k=1 k=−1 and Using the fractions decomposition into prime factors, we get T−[m,n]= ∞ n n 1 n 1 1 1 1 − 2 1 1 m + 1 (−1)n+1 [(m−1)/2] (−1)k (m − 1)! ∑ = ∑ − ∑ + ∑ . (30) ∑ (π n)m−2k−1+ k (k − n) n k n k n k πm+1 m k=−∞ k=1 k=1 k=1 n n k=0 (m − 2k − 1)! k6=0,k6=n (−1)[(m−1)/2] (m − 1)! As a result, we get + m 2[(m − 1)/2] − m + 2 , (39) n ∞ 1 1 2 ∑ = . (31) where n ∈ N and [z] is the integer part of the value z. k=−∞ k (k − n) n n k6=0,k6=n

For n = 0, we have 6 Integral identities with 1F2 functions ∞ 1 ∞ 1 π2 π2 Let us consider the generalized hypergeometric function ∑ 2 = 2 ∑ 2 = 2 = . (32) k=−∞ k k=1 k 6 3 in the form k6=0 1 x2 1F2 α;β,1 ± ;− , (40) 2 4 where α > 0, β > 0 and x ∈ R. Remark 3. The identities (20),(22),(23) with α = β = 2 have the forms Proposition 5. The following identities with the ∞ + 9 generalized hypergeometric functions (40) hold ∑ T+[2,k]T+[2,k − n]= T+[4,n], (33) k=−∞ 5 +∞ 1 y2 +∞ dy 1F2 α1;β1, ;− 9 Z−∞ 2 4 ∑ k T−[2,k]T+[2,k − n]= T−[4,n], (34) 2 k=−∞ 5 1 (x − y) 1F2 α2;β2, ;− = g1(α1,α2,β1,β2) +∞ 9 2 4 ∑ (k − n)T+[2,k]T−[2,k − n]= T−[4,n], (35) 2 ∞ 5 1 3 1 x k=− 1F2 α1 + α2 − ;β1 + β2 − , ;− , (41) 2 2 2 4 where T±[2,n] are presented in (24),(25), and the functions T±[4,n] are deﬁned by the equations where π2 2 5 7 3 n 2π Γ (β1)Γ (β1) T−[4,n]= 1F2 ; , ;− = g1(α1,α2,β1,β2) := 2 2 2 4 Γ (α1)Γ (α2)Γ (β1 − α1) n n 2 5 6(−1) (−1) π Γ (β1 + β2 − α1 − α2 − 1)Γ (α1 + α2 − 1/2) − 4 3 − (n 6= 0), (36) , (42) π n n n Γ (β2 − α2)Γ (β1 + β2 − 3/2)

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and β1 > α1 > 0, β2 > α2 > 0, Proposition 7. The following identities with the β1 + β2 − 3/2 > α1 + α2 − 1/2 > 0. functions (40) hold

Proof. To prove identity (41), the Fourier transform is +∞ 2 α β 3 y applied to equation (41). Using equation (5) of Section dy x 1F2 1; 1, ;− Z−∞ 2 4 1.14 of [13], which has the form 1 (x − y)2 1F2 α2;β2, ;− = g1(α1,α2,β1,β2) ∞ 1 x2 2 4 dx cos(kx) 1F2 α;β, ;− = Z0 2 4 1 3 3 x2 1F2 α1 + α2 − ;β1 + β2 − , ;− , (49) π Γ (β) α β α 2 2 2 4 k2 −1 (1 − k2) − −1 (43) Γ (α)Γ (β − α) where g1(α1,α2,β1,β2) is deﬁned by (42), and for 0 < k < 1, β1 > α1 > 1/2, β2 > α2 > 0, β1 + β2 − 3/2 > α1 + α2 − 1/2 > 1/2. ∞ 1 x2 dx cos(kx) 1F2 α;β, ;− = 0 (44) Z0 2 4 Proof. To prove identity (49), the Fourier transform is applied to equation (49). Using equations (43) and (47), for 1 < k < ∞, where β > α > 0, we obtain the identity. we obtain the identity. Remark 5. The suggested identities for special type Proposition 6. The following identities with the of the generalized hypergeometric functions 1F2(a;b,c;z) generalized hypergeometric functions (40) hold can be used in fractional calculus and fractional dynamics of media and systems with power-law nonlocality. An +∞ 3 y2 dyx(x − y) 1F2 α1;β1, ;− importance of these functions for fractional dynamics is Z−∞ 2 4 based on the power-law form of the Fourier transform of 3 (x − y)2 considered hypergeometric functions. 1F2 α2;β2, ;− = g2(α1,α2,β1,β2) 2 4 3 5 1 x2 1F2 α1 + α2 − ;β1 + β2 − , ;− , (45) References 2 2 2 4 [1] W.N. Bailey, Generalised Hypergeometric Series, Cambridge where University Press, 1935. π Γ β Γ β [2] L.J. Slater, Generalized Hypergeometric Functions, α α β β 2 ( 1) ( 1) g2( 1, 2, 1, 2) := −Γ α Γ α Γ β α Cambridge University Press, 1966. ( 1) ( 2) ( 1 − 1) [3] A.M. Mathai and R.K. Saxena, Generalized Hypergeometric Γ (β1 + β2 − α1 − α2 − 1)Γ (α1 + α2 − 3/2) Functions with Applications in Statistics and Physical , (46) Sciences, Springer, Berlin, Heidelberg, 1973. Γ (β2 − α2)Γ (β1 + β2 − 5/2) [4] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional and β1 > α1 > 1/2, β2 > α2 > 1/2, Integrals and Derivatives Theory and Applications, Gordon β1 + β2 − 5/2 > α1 + α2 − 3/2 > 1/2. and Breach, New York, 1993. [5] V. Kiryakova, Generalized Fractional Calculus and Proof. To prove identity (45), the Fourier transform is Applications, Longman, Harlow and Wiley, New York, applied to equation (45). Using equation (7) of Section 1994. 2.14 of [13], which has the form [6] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, ∞ 3 x2 Amsterdam, 2006. dx sin(kx) x 1F2 α;β, ;− = [7] H.M. Srivastava, Special Functions in Fractional Calculus Z0 2 4 and Related Fractional Differintegral Equations, World π Γ (β) α β α Scientiﬁc, Singapore, 2014. k2 −2 (1 − k2) − −1 (47) Γ (α)Γ (β − α) [8] J. Tenreiro Machado, V. Kiryakova and F. Mainardi, Communications in Nonlinear Science and Numerical for 0 < k < 1, and Simulation. 16, 1140-1153 (2011). [9] V.E. Tarasov, Fractional Dynamics: Applications of ∞ 2 3 x Fractional Calculus to Dynamics of Particles, Fields dx sin(kx) x 1F2 α;β, ;− = 0 (48) Z0 2 4 and Media, Springer, New York, 2011. [10] V.E. Tarasov, Advances in High Energy Physics. 2014, for 1 < k < ∞, where β > α > 1/2, we obtain the 957863 (2014). identity. [11] V.E. Tarasov, Journal of Statistical Mechanics: Theory and Experiment. 2014, P09036 (2014).

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[12] V.E. Tarasov, Physica A: Statistical Mechanics and its Vasily Tarasov received Applications. 421, 330-342. (2015). his Ph.D. in Theoretical [13] A. Erdelyi, Tables of integral transforms. Volume 1, Physics from the Lomonosov McGraw-Hill, New York, 1954. Moscow State University [14] A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, in 1995. He was a Research Integrals and Series, Vol. 1: Elementary Functions, Gordon Associate at the Skobeltsyn and Breach, New York, 1986. Institute of Nuclear Physics, [15] A. Erdelyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Moscow State University for Higher Transcendental Functions. Volume 1, Krieeger, six years, and then became a Melbourne, Florida, 1981. Senior Research Associate at [16] A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, the Skobeltsyn Institute of Nuclear Physics, Lomonosov Integrals and Series, Vol. 2: Special Functions, Gordon and Moscow State University. Vasily is an Associate Breach, New York, 1990. [17] A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Professor at the Applied Mathematics and Physics Integrals and Series, Vol. 3: More Special Functions, Gordon Department of Moscow Aviation Institute. He is a and Breach, New York, 1989. member of Organizing Committee of International Workshop on High Energy Physics and Quantum Field Theory. Vasily E. Tarasov was Senior Visitor at Courant Institute of Mathematical Sciences, New York University in 2005-2009. V.E. Tarasov is an Associate Editor of ”Communications in Nonlinear Science and Numerical Simulations” (Elsevier); ”Journal of Applied Nonlinear Dynamics” (LH Scientific Publishing). V.E. Tarasov is a member of Editorial Boards of the journals: ”Fractional Differential Calculus” (Ele-Math); ”International Journal of Applied and Computational Mathematics” (Springer); ”Discontinuity, Nonlinearity, and Complexity” (LH Scientific Publishing); ”Physics International” (Science Publications). He has published about 160 scientific works, among which 8 books, 4 chapters in books and about 140 papers in refereed journals.

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