The Generalized Hypergeometric Function As the Meijer G-Function

Home , Hypergeometric function, Power series

Analysis 2016; 36 (1):1–14

Research Article

Anatoly A. Kilbas, Ram K. Saxena, Megumi Saigo and Juan J. Trujillo* The generalized hypergeometric function as the Meijer G-function

Abstract: In this paper, we use the representation of the generalized hypergeometric function p Fq in terms of the known Meijer G-function to extend the range of parameters of such a special function so it be convergent. Also, we establish the corresponding series representation for such an extension. In particular, we extend the hypergeometric functions 2F1 and 3F2 from z 1 to z 1. Moreover, we obtain asymptotic formulas to estimate the extended p Fq at innity. | | < | | > Keywords: Generalized hypergeometric function, Mellin–Barnes integral, Meijer G-function, generalized hypergeometric series, Gauss hypergeometric function, series representations, asymptotic estimates

MSC 2010: 33C60, 33C20, 33C05, 41A60

DOI: 10.1515/anly-2015-5001 Received July 5, 2014; accepted March 20, 2015

1 Introduction

The generalized hypergeometric function p Fq a1,..., ap; b1,..., bq; z is dened by the generalized hypergeometric series (see [1, 4.1(1)]) [ ] p ai k +∞ zk F z F a ,..., a ; b ,..., b ; z i=1 , (1.1) p q p q 1 p 1 q q k! k=0 I( ) bj k [ ] ≡ [ ] = H j=1 I( ) where z , p, q 0 0, 1, 2,... and ai , bj , i 1,..., p, j 1,..., q. Here, is the set of complex numbers. An empty product in (1.1), if it occurs, is taken to be one and d k, d , k 0 is the Pochham- mer symbol∈ ℂ (see [1,∈ 1.21(5)])ℕ = { } ∈ ℂ = = ℂ ( ) ∈ ℂ ∈ ℕ d 0 1, d 0,

d k d d 1 d k 1 , k 1, 2,... . .( ) = ̸= The generalized hypergeometric> series includes many elementary and special functions. In particular, F( ) = ( + ) ⋅ ⋅ ⋅ ( + − ) ∈ ℕ = { } when p q 0, (1.1) coincides with the exponential function

+∞ zk = = F ; ; z ez . (1.2) 0 0 k! k=0 [− − ] = ≡ H

Anatoly A. Kilbas: Department of Mathematics and Mechanics, Belarusian State University, Minsk 220050, Belarus Ram K. Saxena: Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur 342011, India, e-mail: [email protected] Megumi Saigo: Department of Applied Mathematics, Fukuoka University, Fukuoka 814-0180, Japan, e-mail: [email protected] *Corresponding author: Juan J. Trujillo: Departamento de Análisis Matemático, Universidad de La Laguna, La Laguna, Tenerife 38271, Spain, e-mail: [email protected] 2 Ë A. A. Kilbas et al., The generalized hypergeometric function as the Meijer G-function

If p 2 and q 1, then (1.1) gives the Gauss hypergeometric series (see [1, 2.1(2)])

= = +∞ a a zk F a , a ; b ; z 1 k 2 k , z 0, (1.3) 2 1 1 2 1 b k! k=0 1 k ( ) ( ) [ ] = H | | < while for p 3 and q 2, (1.1) yields the hypergeometric( series)

= = +∞ a a a zk F a , a , a ; b , b ; z 1 k 2 k 3 k , z 0. (1.4) 3 2 1 2 3 1 2 b b k! k=0 1 k 2 k ( ) ( ) ( ) [ ] = H | | < It is known (see [1, Section 4.1]) that the generalized hypergeometric( ) ( ) series dened by the series in (1.1) is absolutely convergent for all nite z if p q 1 and for z 1 if p q 1. In particular, the series in (1.3) and (1.4) are absolutely convergent for z 1. It is also known (see [1, 5.6(1)], [6,∈ ℂ 7.2.12])< that+ if bj |l,| i< 1,...,= q+, l 0, then the generalized hypergeometric series (1.1) can be represented| | in< terms of the Mellin–Barnes integral (see [1, Section 1.19]) of the form ̸=− = ∈ ℕ

q p Γ bj Γ s Γ ai s j=1 1 i=1 −s p Fq z p Fq a1,..., ap; b1,..., bq; z p q z ds, z 0, (1.5) I ( ) 2πi ( ) I ( − ) Γ ai L Γ bj s [ ] ≡ [ ] = i=1 X j=1 (− ) ̸= I ( ) I ( − ) for bj 0, 1, 2,... , j 1,..., q, bj 0, 1, 2,... , j 1,..., p, and with the specially chosen contour L. Such a formula leads to the representation of (1.1) as the Meijer G-function (see Section 3). In particular, the̸= hypergeometric− − series= (1.3) and (1.4)̸= − have− the representations=

Γ b1 1 Γ s Γ a1 s Γ a2 s −s 2F1 a1, a2; b1; z z ds (1.6) Γ a1 Γ a2 2πi Γ b1 s ( ) L ( ) ( − ) ( − ) [ ] = X (− ) ( ) ( ) ( − ) with b1 l, l 0, and

̸=− ∈ ℕ Γ b1 Γ b2 1 Γ s Γ a1 s Γ a2 s Γ a3 s −s 3F2 a1, a2, a3; b1, b2; z z ds (1.7) Γ a1 Γ a2 Γ a3 2πi Γ b1 s Γ b2 s ( ) ( ) L ( ) ( − ) ( − ) ( − ) [ ] = X (− ) ( ) ( ) ( ) ( − ) ( − ) with bj l, j 1, 2, l 0, respectively. In this paper, we use the representation (1.5) as the denition of the generalized hypergeometric function p Fq a̸=−1,...,= ap; b1∈,...,ℕ bq; z . Changing the contour of integration L, we extend the range of parameters of such a function to all nite z 0 for p q 1 and z 1 for p q 1, and we establish the series representations[ for such an extension.] In particular, we extend the hypergeometric functions 2F1 a1, a2; b1; z and 3F2 a1, a2, a3; b1, b2; z from̸= z 1> to+z 1.| The| > obtained= formulas+ yield asymptotic estimates for p Fq a1,..., ap; b1,..., bq; z at innity, especially for 2F1 a1, a2; b1; z and 3F2 a1, a2, a[3; b1, b2; z . ] It should[ be noted that the] above| | approach< | | allows> us to make a meaning of the hypergeometric function p F[q a1,..., ap; b1,..., bq;]z when p q 1. It is also known[ that another] approach,[ based on the] MacRobert E-function (see, for example, [1, Section 5.2]), gives such a meaning. The[ paper is organized as follows.] Section> + 2 contains results giving conditions for the existence of the generalized hypergeometric function p Fq a1,..., ap; b1,..., bq; z dened by the Mellin–Barnes integral (1.5). A representation of this function as the Meijer G-function is given in Section 3. The series representations of such a generalized hypergeometric[ function are established] in Section 4. Similar results for the Gauss hypergeometric function 2F1 a1, a2; b1; z and for the generalized hypergeometric functions 3F2 a1, a2, a3; b1, b2; z and 3F3 a1, a2, a3; b1, b2, b3; z are presented in Section 5. [ ] [ ] [ ] A. A. Kilbas et al., The generalized hypergeometric function as the Meijer G-function Ë 3

2 The generalized hypergeometric function as the Mellin–Barnes integral

In this section, we give conditions for the existence of the generalized hypergeometric function p Fq z dened by the Mellin–Barnes integral of the form (1.5). These conditions will be dierent for the innite contour L, which has one of the following forms. [ ]

(i) L L−∞ is a left loop starting at and ending at , enclosing all the poles of Γ s . (ii) L L+∞ is a right loop starting at and ending at , enclosing all the poles of Γ aj s , j 1,2,..., p, situated= in a horizontal strip starting−∞ at the point −∞ iφ1 and terminating at the( point) iφ2 with = φ1 φ2 . +∞ +∞ ( − ) = (iii) L Liã∞ is a contour starting at the point i and+∞ terminating+ at the point i , where+∞ + . We−∞ suppose< < that< +a∞i , bj , i 1,..., p, j 1,..., q, be such that bj l, j 1,..., q, l 0, and the poles= ã − ∞ ã + ∞ ã ∈ ℝ ∈ ℂ = cl = l, l 0, ̸=− = ∈ ℕ (2.1) of the gamma function Γ s and the poles = − ∈ ℕ ( ) aik ai k, i 1,..., p, k 0, (2.2) of the gamma functions Γ ai s , i 1,...,= +p, are separated,= that∈ is,ℕ

( − ) a=i k l, i 1,..., p, k, l 0. (2.3)

We also suppose that the poles aik in (2.2)+ are̸=− simple,= that is, ∈ ℕ

ai k aj m, i j, i, j 1,..., p, k, m 0. (2.4)

We shall use the notation + ̸= + ̸= = ∈ ℕ q p p q µ b a . j i 2 j=1 i=1 − When L L−∞, the existence of the generalized= H − H hypergeometric+ function p Fq z dened by the Mellin– Barnes integral (1.5) is given by the following result. = [ ] Theorem 2.1. Let ai , bj , j 1,..., p , j 1,..., q, be such that bj l, i 1,..., q, l 0, and let the conditions in (2.3) and (2.4) be satised. Let one of the conditions ∈ ℂ = = ̸=− = ∈ ℕ q 1 p, z 0, (2.5) q 1 p, 0 z 1, (2.6) + > ̸= 1 q 1 p, z 1, Re µ , (2.7) + = < | | < 2 hold. Then, the generalized hypergeometric+ function= | p|F=q z dened( ) > by the Mellin–Barnes integral (1.5) exists, where the path of integration L L−∞ separates all poles cl in (2.1) to the left and all poles aik in (2.2) to the right. [ ] = Proof. Let p Γ s Γ ai s G1,p i=1 p,q+1 s q . (2.8) ( ) I ( − ) Γ bj s ( ) = j=1 In accordance with the formula I ( − ) 1 Γ z α 2π 1/2zz+α−1/2e−z 1 O (2.9) z ( + ) = ( ) + 4 Ë A. A. Kilbas et al., The generalized hypergeometric function as the Meijer G-function as z and for α bounded, which is connected with the well-known Stirling formula for the Gamma function [1, 1.18(2)], the asymptotic relation at innity (see [2, (1.2.1)]) | | → +∞ Γ x iy 2π 1/2 x x−1/2e−x−π(1−sign(x))y/2, x, y , (2.10) holds as x . By applying| ( + this)| ∼ formula,( ) | a direct| calculation leads to the∈ asymptoticℝ estimate e (q−p+1)|t| | | → +∞ G1,p t iσ A t −(Re(µ)+1/2), t, σ , p,q+1 t ᐈ ᐈ 1,p ᐈ ( + )ᐈ ∼ | | ∈ ℝ for the function Gp,q+1 s inᐈ (2.8) as t ᐈ , where| |

q p ( ) (→p−q −+1∞)/2 A 2π exp Re bj Re ai πσ . j=1 i=1 = ( ) ¤H ( ) − H ( ) − ¥ Since z −(t+iσ) z −t eσ arg(−z)e−i(t arg(−z)+σ log |z|), (2.11) then, in accordance with the known convergence principle for improper integrals, the integral in (1.5) is ab- (− ) = | | solutely convergent, provided that either one of the conditions in (2.5), (2.6) and (2.7) is satised. This completes the proof of the theorem.

Corollary 2.2. Let a1, a2, b1 be such that b1 l, l 0, and let the conditions in (2.3) and (2.4) with p 2 be satised. Let either one of the conditions ∈ ℂ ̸=− ∈ ℕ = 0 z 1 (2.12) or < | | < z 1, Re b1 a1 a2 0, (2.13) hold. Then, the hypergeometric function F a , a ; b ; z dened by the Mellin–Barnes integral (1.6) exists, | |2=1 1 2( 1 − − ) > where the path of integration L L−∞ separates all poles cl in (2.1) to the left and all poles aik, i 1, 2, in (2.2) to the right. [ ] = = Corollary 2.3. Let a1, a2, a3, b1, b2 be such that bj l, j 1, 2, l 0, and let the conditions in (2.3) and (2.4) with p 3 be satised. Let either one of the conditions (2.12) or ∈ ℂ ̸=− = ∈ ℕ = z 1, Re b1 b2 a1 a2 a3 0, (2.14) hold. Then, the hypergeometric function| | = 3F2 a1(, a2+, a3;−b1, b−2; z −dened) > by the Mellin–Barnes integral (1.7) exists, where the path of integration L L−∞ separates all poles cl in (2.1) to the left and all poles aik, i 1, 2, 3, in (2.2) to the right. [ ] = = The next result yields the existence of the generalized hypergeometric function p Fq z dened by the Mellin– Barnes integral (1.5) for L L+∞. ( ) Theorem 2.4. Let a , b , i 1,..., p, j 1,..., q, be such that b l, i 1,..., q, l , and let the i j = j 0 conditions in (2.3) and (2.4) be satised. Let one of the conditions ∈ ℂ = = ̸=− = ∈ ℕ q 1 p, z 0, (2.15) q 1 p, z 1, (2.16) + < ̸= 1 q 1 p, z 1, Re µ , (2.17) + = | | > 2 hold. Then, the generalized hypergeometric+ function= | p|F=q z dened( ) > by the Mellin–Barnes integral (1.5) exists, where the path of integration L L+∞ separates all poles cl in (2.1) to the left and all poles aik in (2.2) to the right. [ ] = A. A. Kilbas et al., The generalized hypergeometric function as the Meijer G-function Ë 5

Proof. Using (2.10), we deduce the asymptotic estimate

e −(q−p+1)t G1,p t iσ B t −(Re(µ)+1/2), t, σ , (2.18) p,q+1 t ᐈ ᐈ G1,p ᐈ ( + )ᐈ ∼ | | ∈ ℝ for the function p,q+1 s ᐈin (2.8) as t ᐈ , where| | q p ( ()p+1−q)/2 (p−q)σπ→ +∞ B 2π e exp Re bj π Im bj Re ai π Im ai . j=1 i=1 = ( ) ¤H( ( ) + ( )) − H( ( ) + ( ))¥ According to (2.18), (2.11) and the known convergence principle for improper integrals, the integral in the right-hand side of (1.5) is absolutely convergent, provided that either one of the conditions in (2.15), (2.16) and (2.17) is satised. Thus, the theorem is proved.

Corollary 2.5. Let a1, a2, b1 be such that b1 l, l 0, and let the conditions in (2.3) and (2.4) with p 2 be satised. Let either one of the conditions ∈ ℂ ̸=− ∈ ℕ = z 1 (2.19) or (2.13) hold. Then, the hypergeometric function 2F|1 |a>1, a2; b1; z dened by the Mellin–Barnes integral (1.6) exists, where the path of integration L L+∞ separates all poles cl in (2.1) to the left and all poles aik, i 1, 2, in (2.2) to the right. [ ] = = Corollary 2.6. Let a1, a2, a3, b1, b2 be such that bj l, j 1, 2, l 0, and let the conditions in (2.3) and (2.4) with p 3 be satised. Let either one of the conditions (2.19) or (2.14) hold. Then, the hypergeometric function 3F2 a1, a2, a3; b1, b2; z dened∈ ℂ by the Mellin–Barnes̸=− integral= (1.7)∈ ℕ exists, where the path of integration L L+∞ separates= all poles cl in (2.1) to the left and all poles aik, i 1, 2, 3, in (2.2) to the right. [ ] Finally, we deduce the existence of the generalized hypergeometric function F z dened by the Mellin– = = p q Barnes integral (1.5) for L Liã∞. [ ] Theorem 2.7. Let a , b , i 1,..., p, j 1,..., q, be such that b l, j 1,..., q, l , and let the i j = j 0 conditions in (2.3) and (2.4) be satised. Let one of the conditions ∈ ℂ = = ̸=− = ∈ ℕ p q 1 π p 1 q, arg z , z 0, (2.20) 2 1 p 1 q, Re µ 2 ( −, z +is real) negative, (2.21) + > | (− )| < 2 ̸= hold. Then, the generalized hypergeometric+ = function( ) > pãFq+ z dened by the Mellin–Barnes integral (1.5) exists, where the path of integration L Liã∞ separates all poles cl in (2.1) to the left and all poles aik in (2.2) to the right. [ ] = Proof. Using the asymptotic relation at innity (see [2, (1.2.2)])

1,p (q−p+1)σ−Re(µ)−1/2 −(p−q+1)|t|π/2 | | → +∞ Gp,q+1 σ it C t e , σ, t , (2.22) for the function G1,p ᐈs in (2.8) as ᐈt , where p,q+1 ᐈ ( + )ᐈ ∼ | | ∈ ℝ π (C) 2π (p−q+1|)/|2→e(p +−∞q−1)σ−(p−q)/2 exp Re µ sign t Im µ . 2 By (2.22), (2.11) and the= known( ) convergence principle for improper ( ) − integrals,( ) the( ) integral in the right-hand side of (1.5) is absolutely convergent, provided that either one of the conditions in (2.20) and (2.21) is satised. This completes the proof of the theorem. 6 Ë A. A. Kilbas et al., The generalized hypergeometric function as the Meijer G-function

Corollary 2.8. Let a1, a2, b1 be such that b1 l, l 0, let the conditions in (2.3) and (2.4) with p 2 be satised and let z be such that arg z π. Then, the hypergeometric function 2F1 a1, a2; b1; z dened by the Mellin–Barnes integral∈ ℂ (1.6) exists, where̸=− the∈ ℕ path of integration L Liã∞ separates all poles=cl in (2.1) to the left and∈ allℂ poles aik, i 1| , 2,(− in )|(2.2)< to the right. [ ] = Corollary 2.9. Let a , a , a , b , b be such that b l, i 1, 2, l , let the conditions in (2.3) 1 2 3 1 2= i 0 and (2.4) with p 3 be satised and let z be such that arg z π. Then, the hypergeometric function 3F2 a1, a2, a3; b1, b2; z dened∈ byℂ the Mellin–Barnes integral̸=− =(1.7) exists,∈ ℕ where the path of integration L Liã∞ separates= all poles cl in (2.1) to the∈ leftℂ and all poles a|ik, i(− 1)|,<2, 3, in (2.2) to the right. [ ] Remark 2.10. Theorem 2.4 allows us to dene the generalized hypergeometric function F z dened = = p q by (1.5) with L L+∞ for the range of parameters p, q 0 such that p q 1 for all z and p q 1 for z 1. This representation can be considered as an extension of the generalized hypergeometric[ ] function dened by the series= in (1.1) from the usual range of the∈ ℕ parameters and> the+ variable z,∈ thatℂ is, p = q + 1 for| all| >z and p q 1 for z 1. Similarly, in the case p q 1, the representation in (1.5) can be considered as the extension of the generalized hypergeometric function (1.1) from z 1 to z 1. < + ∈ ℂ = + | | < = + Remark 2.11. The approach described above has a meaning for the hypergeometric function F a ,..., a ; | | < p| |q> 1 p b1,..., bq; z when p q 1 is based on the Mellin–Barnes integral representation (1.5). It is also known that another approach gives such a meaning to p Fq a1,..., ap; b1,..., bq; z for p q 1, based[ on the introduction of] the so-called> + MacRobert E-function (see [1, Section 5.2]). [ ] > + Remark 2.12. The representation (1.6) for the Gauss hypergeometric function 2F1 a1, a2; b1; z , dened by the hypergeometric series (1.2), is well known (see [1, 2.1(15)]). [ ]

3 The generalized hypergeometric function as the Meijer G-function

We now apply the results in Theorem 2.1, Theorem 2.4 and Theorem 2.7 of Section 2 to represent the generalized hypergeometric function p Fq a1,..., ap; b1,..., bq; z dened by (1.5) with ai , bj , i 1,..., p, m,n j 1,..., q, as a special case of the Meijer G-function Gp,q z (see, for example, [1, Section 5.3]). This function, for m, n, p, q 0 such that 0[ n p, 1 m q, and for] ai , bj , is dened by means∈ ℂ of= a Mellin– Barnes-type= integral as ( ) ∈ ℕ ≤ ≤ ≤ ≤ ∈ ℂ m,n m,n a1,..., ap 1 −s Gp,q z Gp,q z Θ s z ds, (3.1) b1,..., bq 2πi ᐈ L ᐈ ᐈ where ( ) ≡ ᐈ ¡ = X ( ) m ᐈ n Γ bj s Γ 1 ai s j=1 i=1 Θ s . (3.2) pI ( + ) Iq ( − − ) Γ ai s Γ 1 bj s ( ) = i=n+1 j=m+1 Here, I ( + ) I ( − − ) z−s exp s log z i arg z , z 0, i 1, where log z represents the natural logarithm of z and arg z is not necessarily the principal value. An empty = (− ( | | + )) ̸= = $− product in (3.2), if it occurs, is taken to be one and the poles | | | | bjl bj l , j 1,..., m, l 0, (3.3) of the gamma functions Γ b s and the poles j = −( + ) = ∈ ℕ a 1 a k, i 1,..., n, k , (3.4) ( + ) ik i 0

= − + = ∈ ℕ A. A. Kilbas et al., The generalized hypergeometric function as the Meijer G-function Ë 7

of the gamma functions Γ 1 ai s do not coincide, that is, b l a k 1, i 1,..., n, j 1,..., m, l, k . (j − −i ) 0 The contour L in (3.1) is one of the above contours L L , L L or L L , which separates all + ̸= − − = −=∞ +∞ ∈ ℕ iã∞ poles bjl in (3.3) to the left and all poles aik in (3.4) to the right of L. The theory of the Meijer G-function may be found in [1,= Sections= 5.3–5.6], in= [3] and in [6, §8.2]. We also note that the Meijer G-function is a special case of the H-function, that is,

m,n a1,..., ap m,n a1, 1 ,..., ap , 1 Gp,q z Hp,q z . (3.5) b1,..., bq b1, 1 ,..., bq , 1 ᐈ ᐈ ᐈ ᐈ ( ) ( ) A detailed and comprehensive account ᐈ of the H-function¡ ≡ is availableᐈ in the monographs¡ by Mathai [4], Mathai ᐈ ᐈ ( ) ( ) and Saxena [5], Srivastava, Guptaᐈ and Goyal [10], Prudnikov,ᐈ Brychkov and Marichev [6, §8.3], and Kilbas and Saigo [2, Chapters 1–2]. According to (1.5), (3.1)–(3.2) and (3.5), we obtain the representation of the generalized hypergeometric function p Fq a1,..., ap; b1,..., bq; z as a G-function of the form q [ ] Γ bj j=1 1,p 1 a1,..., 1 ap p Fq a1,..., ap; b1,..., bq; z G z . (3.6) Ip ( ) p,q+1 0, 1 b ,..., 1 b ᐈ 1 q Γ ai ᐈ − − ᐈ [ ] = i=1 − ᐈ ¡ ᐈ − − From Theorem 2.1, Theorem 2.4 and Theorem 2.7,I we( deduce) the conditions for these representations.

Theorem 3.1. Let p, q 0, let ai , bj , i 1,..., p, j 1,..., q, be such that bj l, j 1,..., q, l 0, let the conditions in (2.3) and (2.4) be satised, and let . Let L be the contour which separates all poles cl in (2.1) to the left and all∈ ℕ poles aik in (2.2)∈ ℂ to= the right and= let one of the following conditions̸=− = be valid. ∈ ℕ (i) L L−∞ and either one of the conditions in (2.5), (2.6)ã ∈ ℝor (2.7) holds. (ii) L L+∞ and either one of the conditions in (2.15), (2.16) or 2.17 holds. (iii) L = Liã∞ and either one of the conditions in (2.20) or (2.21) holds. Then, the= generalized hypergeometric function p Fq a1,..., ap; b1,..., bq; z dened by (1.5) is represented as a G-function= by (3.6). [ ] Corollary 3.2. Let a1, a2, b1 be such that b1 l, l 0, let the conditions in (2.3) and (2.4) with p 2 be satised, and let . Let L be the contour which separates all poles cl in (2.1) to the left and all poles aik, i 1, 2, in (2.2) to the right and∈ ℂ let one of the following̸=− conditions∈ ℕ be valid. = (i) L L−∞ and eitherã ∈ ℝ one of the conditions in (2.12) or (2.13) holds. (ii)= L L+∞ and either one of the conditions in (2.19) or (2.13) holds. (iii) L = Liã∞ and z is such that arg z π. Then, the= Gauss hypergeometric function 2F1 a1, a2; b1; z dened by (1.6) is represented as a G-function as = ∈ ℂ | (− )| < Γ b1 1,2 1 a1, 1 a2 2F1 a1, a2; b1; z [ G] 2,2 z . (3.7) Γ a1 Γ a2 0, 1 b1 ᐈ ( ) ᐈ − − Corollary 3.3. Let a , a , a ,[ b , b be] = such that b l, j− 1ᐈ , 2, l , let¡ the conditions in (2.3) and 1 2 3 1 2 ( ) ( j) ᐈ − 0 (2.4) with p 3 be satised, and let . Let L be the contour which separates all poles cl in (2.1) to the left and all poles aik, i 1, 2, 3, in (2.2) to∈ theℂ right and let one̸=− of the= following∈ ℕ conditions be valid. (i) L L−∞=and either one of the conditionsã ∈ ℝ in (2.12) or (2.14) holds. (ii) L L+∞ and either= one of the conditions in (2.19) or (2.14) holds. (iii) L = Liã∞ and z is such that arg z π. Then, the= generalized hypergeometric function 3F2 a1, a2, a3; b1, b2; z dened by (1.7) is represented as a G-function= as ∈ ℂ | (− )| < [ ] Γ b1 Γ b2 1,3 1 a1, 1 a2, 1 a3 3F2 a1, a2, a3; b1, b2; z G3,3 z . Γ a1 Γ a2 Γ a3 0, 1 b , 1 b ᐈ 1 2 ( ) ( ) ᐈ − − − ᐈ [ ] = − ᐈ ¡ ( ) ( ) ( ) ᐈ − − 8 Ë A. A. Kilbas et al., The generalized hypergeometric function as the Meijer G-function

Remark 3.4. The representation of p Fq a1,..., ap; b1,..., bq; z in the form (3.6) is well known (see, for example, [1, 5.6(1)] and [6, 8.4.51]). The conditions (2.5) and (2.6) for such a representation given in The- orem 3.1(i) may also be known because[ they coincide, except for] the point z 0, with the well-known conditions for convergence of the generalized hypergeometric series (1.1) (see [1, Section 4.1]). The condition (2.7) in Theorem 3.1(i) as well as conditions (2.15)–(2.17) in Theorem 3.1(ii)= and conditions (2.20)–(2.21) in Theorem 3.1(iii) have not appeared before.

Remark 3.5. Theorem 2.4 and Theorem 3.1(ii) allow us to dene the generalized hypergeometric function p Fq a1,..., ap; b1,..., bq; z by (1.5) and (3.6) for the parameters p, q and the variable z such that p q 1 for all complex z 0 and p q 1 for z 1. These representations can be considered as an extension[ of the generalized hypergeometric] function dened by the series in (1.1) from the usual∈ ℂ range of the parameters> + p, q and the variable̸= z,= that+ is, p | q| > 1 for all z and p q 1 for z 1. Similarly, in the case p q 1, the relations (1.5) and (3.6) can be considered as an extension of the generalized hypergeometric function (1.1) from z 1 to z 1.< + ∈ ℂ = + | | < = + Remark 3.6. The representation (3.7) for the Gauss hypergeometric function F a , a ; b ; z is well known | | < | | > 2 1 1 2 1 (see, for example, [2, (2.9.15)]). [ ]

4 Series representations of the generalized hypergeometric function

In this section, we prove the series representation for the generalized hypergeometric function pFq a1,...,ap; b1,..., bq; z dened by (1.5) and (3.6). These representations are dierent for L−∞ and L+∞. From Theo- rem 2.1 we deduce the rst result, which yields the series representation of the generalized hypergeometric[ function at zero.]

Theorem 4.1. Let p, q 0, let ai , bj , i 1,..., p, j 1,..., q, be such that bj l, j 1,..., q, l 0, and let the conditions in (2.3) and (2.4) be satised. Let either one of the conditions in (2.5), (2.6) or (2.7) hold and let L−∞ be the contour∈ ℕ which separates∈ ℂ all= poles cl in=(2.1) to the left and all poles̸=−aik in= (2.2) to the∈ right.ℕ Then, the generalized hypergeometric function p Fq a1,..., ap; b1,..., bq; z dened by (1.5) with L L−∞ has the power series expansion (1.1), that is, [ ] = p ai k +∞ zk F a ,..., a ; b ,..., b ; z i=1 . (4.1) p q 1 p 1 q q k! k=0 I( ) bj k [ ] = H j=1 I( ) Proof. Using (2.8), we write p Fq z dened by (1.5) in the form

q [ ] Γ bj j=1 1 1,p −s p Fq a1,..., ap; b1,..., bq; z G s z ds. (4.2) Ip ( ) 2πi p,q+1 Γ ai L−∞ [ ] = i=1 X ( )(− ) I ( ) Applying the usual procedure (see, for example, [1, Section 2.3]), we evaluate the above integral as a sum of the residues of the integrand in (4.2) at the simple poles cl in (2.1). Taking into account (2.8) and the residue at z ν of Γ z , namely, 1 ν lim z ν Γ z , ν 0, 1,..., (4.3) = − ( ) z→ν ν! (− ) ( + ) ( ) = = A. A. Kilbas et al., The generalized hypergeometric function as the Meijer G-function Ë 9 we have

q p q p Γ bj Γ s Γ ai s Γ bj Γ ai k +∞ +∞ k j=1 i=1 −s j=1 i=1 z p Fq a1,..., ap; b1,..., bq; z Res z . p s=−k q p q k! I ( ) k=0 ( ) I ( − ) I ( ) k=0 I ( + ) Γ ai Γ bj s Γ ai Γ bj k [ ] = i=1 H ã j=1 (− ) ë = i=1 H j=1 I ( ) I ( − ) I ( ) I ( + ) From here, in accordance with the formula

Γ α k Γ α α k , α , k 0, we get the result in (4.1). ( + ) = ( )( ) ∈ ℂ ∈ ℕ To obtain a series representation of the generalized hypergeometric function at innity, we need a preliminary assertion which is directly veried by using the asymptotic formula (2.10).

Lemma 4.2. Let c and k . Then, the asymptotic estimates

k k ∈ ℂ ∈ ℕΓ c k C kRe(c)−1/2, C 2π 1/2, (4.4) 1 e 1 and | ( + )| ∼ = ( ) k −k Γ c k C kRe(c)−1/2, C 2π 1/2e−π Im(c), (4.5) 2 e 2 hold as k . | ( − )| ∼ = ( ) Using this lemma on the basis of Theorem 2.4, we give the series representation of the generalized hyperge- → +∞ ometric function p Fq z . Theorem 4.3. Let p , q , let a , b , j 1,..., p, j 1,..., q, be such that b l, i 1,..., q, [ ] 0 i j j l 0, and let the conditions in (2.3) and (2.4) be satised. Let either one of the conditions in (2.15), (2.16) or (2.17) hold and let L+∈∞ℕbe the∈ ℕ contour which∈ separatesℂ = all poles=cl in (2.1) to the left and all poles̸=− aik=in (2.2) to the∈ ℕ right. Then, the generalized hypergeometric function p Fq a1,..., ap; b1,..., bq; z dened by (1.5) with L L+∞ has the series expansion [ ] = q p Γ bj Γ ah k Γ ai ah k p +∞ k a +k j=1 i=1,i ̸=h 1 1 h F a ,..., a ; b ,..., b ; z (4.6) p q 1 p 1 q p q k! z I ( ) h=1 k=0 ( + ) I ( − − ) Γ ai Γ bj ah k (− ) [ ] = i=1 H H j=1 − I ( ) I ( − − ) q p Γ bj Γ ah Γ ai ah p a j=1 i=1,i ̸=h 1 h F a , 1 a b ,... p q z q+1 p−1 h h 1 I ( ) h=1 ( ) I ( − ) Γ ai Γ bj ah = i=1 H j=1 − + − I ( ) I ( − ) 1 p−q+1 ..., 1 a b ; 1 a a ,... , 1 a a ; , (4.7) h q h 1 h p z (∗) (− ) + − + − + − where the asterisk in (4.7) indicates the omission of the parameter 1 ah ah, h 1,..., p. Proof. By Theorem 2.4, the representation in (1.5) is valid, where the contour L L separates all poles c + − = +∞ l in (2.1) to the left and all poles ahk ah k, h 1,..., p, k 0, to the right. We evaluate (1.5) as a sum of = = + = ∈ ℕ 10 Ë A. A. Kilbas et al., The generalized hypergeometric function as the Meijer G-function

residues of the integrand in (1.5) at the points ahk, that is,

q p Γ bj Γ s Γ ai s j=1 1 i=1 s p Fq a1,..., ap; b1,..., bq; z p q z ds I ( ) 2πi ( ) I ( − ) Γ ai L+∞ Γ bj s [ ] = i=1 X j=1 (− ) I ( ) I ( − ) q p Γ bj Γ s Γ ai s p +∞ j=1 Res i=1 z s . p s=ahk q I ( ) h=1 k=0 ( ) I ( − ) Γ ai Γ bj s = − i=1 H H ã j=1 (− ) ë I ( ) I ( − ) By (2.4), all poles ahk ah k are simple. Calculating the above residues and taking into account the asymptotic relations k+1 = + 1 Γ ah s , i 1,..., p, k 0, k! s ah k (− ) as s ah k, which follow from( the− ) asymptotic∼ formula (4.3),= we obtain∈ (4.6).ℕ We now prove the convergence of the series( − in− (4.6).) Let us consider the series → + 1 ah +∞ 1 k c , (4.8) z hk z k=0 where − H − p Γ ai ah k k i=1, i ̸=h 1 chk Γ ah k , j 1,..., p. Iq ( − − ) k! Γ bj ah k (− ) = j=1 ( + ) = Using (4.4) and (4.5), we obtain theI asymptotic( − − estimate)

k (q−p+1)k c C k(1+q−p) Re(ah)−Re(µ)−1/2 hk h e as k , where | | ∼ (p−q−1)/2 Ch 2π exp Im µ ah p q π → +∞ and µ is given in (2.3). Now, the convergence of the series in (4.8) follows from the known convergence prin- = ( ) (( ( + ( − ))) ) ciple for the power series, which completes the proof of (4.6). The relation (4.7) is derived from (4.6) by using the formula

k Γ a Γ a k 1 , a , k 0. 1 a k ( ) This completes the proof of the theorem.( − ) = (− ) ∈ ℂ ∈ ℕ ( − ) Corollary 4.4. Let p , q 0, let ai , bj , i 1,..., p, j 1,..., q, be such that bj l, j 1,..., q, l 0, and let the conditions in (2.3) and (2.4) be satised. Let either one of the conditions in (2.15), (2.16) or (2.17) hold and let L+∈∞ℕbe the∈ ℕ contour which∈ separatesℂ = all poles =cl in (2.1) to the left and all poles̸=− aik=in (2.2) to the∈ ℕ right. Then, the generalized hypergeometric function p Fq a1,..., ap; b1,..., bq; z dened by (1.5) with L L+∞ has the asymptotic estimate

q [ p ] = Γ bj Γ ai ah p a j=1 i=1,i ̸=h 1 h 1 F a ,..., a ; b ,..., b ; z Γ a 1 O (4.9) p q 1 p 1 q p h q z z I ( ) h=1 I ( − ) Γ ai Γ bj ah [ ] = i=1 H ( ) j=1 − + I ( ) I ( − ) A. A. Kilbas et al., The generalized hypergeometric function as the Meijer G-function Ë 11 as z . In particular, 1 a∗ p Fq a1,..., ap; b1,..., bq; z O (4.10) → +∞ z as z , where [ ] = ∗ a min Re ah . (4.11) 1 h p → +∞ ≤ ≤ When p q 1, Theorem 4.3 yields the following= result. ( ) Theorem 4.5. Let q , let a , b , i 1,..., q 1, j 1,..., q, be such that b l, j 1,..., q, = + 0 i j j l 0, and let the conditions in (2.3) and (2.4) be satised with p q 1. Let either one of the conditions ∈ ℕ ∈ ℂ = + = ̸=− = ∈ ℕ z 1 = + or | | > q q+1 z 1, Re bj Re ai , j=1 i=1 hold and let L+∞ be the contour which| | separates= H all( poles) > Hcl in ((2.1)) to the left and all poles aik in (2.2) with p q 1 to the right. Then, the generalized hypergeometric function q+1Fq a1,..., aq+1; b1,..., bq; z dened by (1.5) with L L+∞ and p q 1 has the series expansion = + [ ] q q+1 = = + Γ bj Γ ah k Γ ai ah k q+1 +∞ k a +k j=1 i=1,i ̸=h 1 1 h q+1Fq a1,..., aq+1; b1,..., bq; z q+1 q k! z I ( ) h=1 k=0 ( + ) I ( − − ) Γ ai Γ bj ah k (− ) [ ] = i=1 H H j=1 − I ( ) I ( − − ) q q+1 Γ bj Γ ah Γ ai ah q+1 a j=1 i=1,i ̸=h 1 h q+1Fq ah , 1 ah b1,... q+1 q z I ( ) h=1 ( ) I ( − ) Γ ai Γ bj ah = i=1 H j=1 − + − 1 I ...,( ) 1 a Ib ; 1( a− )a ,... , 1 a a ; , (4.12) h q h 1 h q+1 z (∗) + − + − + − where the asterisk in (4.12) indicates the omission of the parameter 1 ah ah, h 1,..., q 1. Corollary 4.6. Let q , let a , b , i 1,..., q 1, j 1,..., q, be such that b l, i 1,..., q, 0 i j + − = j + l 0, and let the conditions in (2.3) and (2.4) be satised with p q 1. Let L+∞ be the contour which separates all poles cl in (2.1)∈ ℕ to the left and∈ ℂ all poles= aik in +(2.2) with= p q 1 to the right. Then,̸=− the generalized= hypergeometric∈ ℕ function p Fq+1 a1,..., aq+1; b1,..., bq; z dened= by+(1.5) with L L+∞ and p q 1 has the asymptotic estimate of the form (4.9) with p q 1 as z , =arg + z π. In particular, (4.10) is valid, where a∗ is given by (4.11) with[ p q 1. ] = = + = + → +∞ | (− )| < Remark 4.7. The formula (4.12) was given in [6, 7.2.3.77]. = +

5 Special cases of the series representations of the generalized hypergeometric function

In this section, we present some special cases of the representations for the generalized hypergeometric functions established in the previous section. We begin with the case p 2 and q 1, when Theorem 4.1 and Theorem 4.3 yield the corresponding result for the Gauss hypergeometric function 2F1 a1, a2; b1; z . = = [ ] 12 Ë A. A. Kilbas et al., The generalized hypergeometric function as the Meijer G-function

Theorem 5.1. Let a1, a2, b1 be such that

b1 l, a∈1ℂ k l, a2 m l, a1 k a2 m, k, l, m 0, (5.1) and let either one of the̸=− conditions+ in̸=−(2.12) or+ (2.13)̸=− hold. Let+ L ̸= L−+∞ be the contour∈ ℕ which separates all poles cl in (2.1) to the left and all poles a1 k and a2 m, k, m 0, to the right. Then, the Gauss hypergeometric function 2F1 a1, a2; b1; z dened by the Mellin–Barnes= integral (1.6) with L L−∞ has the power series representation (1.3). + + ∈ ℕ [ ] = Theorem 5.2. Let a1, a2, b1 be such that the condition in (5.1) are satised and let either one of the conditions in (2.13) or (2.19) hold. Let L L+∞ be the contour which separates all poles cl in (2.1) to the left and all poles a1 k and a2 m, k, m∈ ℂ 0, to the right. Then, for the Gauss hypergeometric function 2F1 a1, a2; b1; z dened by the Mellin–Barnes integral= (1.6) with L L+∞ holds the formula + + ∈ ℕ [ ] a Γ b1 Γ a2 a1 1 1 1 2F1 a1, a2; b1; z = 2F1 a1, 1 a1 b1; 1 a1 a2; Γ a2 Γ b1 a1 z z ( ) ( − ) a [ ] = Γ b1 Γ a1− a2 1 2 + − + − 1 ( ) ( − ) 2F1 a2, 1 a2 b1; 1 a2 a1; . (5.2) Γ a1 Γ b1 a2 z z ( ) ( − ) Corollary 5.3. Let a , a , b be+ such that the conditions− in (5.1) are+ satised.− + Then,− the Gauss hyper- 1 2 1 ( ) ( − ) geometric function 2F1 a1, a2; b1; z has the asymptotic estimate ∈ ℂ a a Γ b1 Γ a2 a1 1 1 1 Γ b1 Γ a1 a2 1 2 1 2F1 a1, a2; b1; z [ ] 1 O 1 O (5.3) Γ a2 Γ b1 a1 z z Γ a1 Γ b1 a2 z z ( ) ( − ) ( ) ( − ) [ ] = − + + − + as z . In particular,( ) ( − ) ( ) ( − ) 1 a∗ F a , a ; b ; z O (5.4) → +∞ 2 1 1 2 1 z as z , where [ ] = ∗ a min Re a1 , Re a2 . → +∞ The formula (5.2) is well known as the analytic continuation of (1.3) to the domain arg z π Remark 5.4. = ( ) ( ) and the relation (5.3) is a very useful formula for nding the asymptotic expansions of hypergeometric functions of two or more variables and generalized elliptic-type integrals. In this connection, the reader| is(− referred)| < to the papers by Saxena and Kalla [7], Saxena, Kalla and Hubbell [8], and Saxena and Pathan [9].

Remark 5.5. The asymptotic relation formula (5.4) is well known (see, for example, [1, 2.3(9)]).

When p 3 and q 2, from Theorem 4.1 and Theorem 4.3 we deduce the result for the generalized hypergeometric function 3F2 a1, a2, a3; b1, b2; z . = = Theorem 5.6. Let a , a , a , b , b be such that 1 [ 2 3 1 2 ]

bj l, ∈aℂi k l, i 1, 2, 3, j 1, 2, k, l 0, (5.5) and ̸=− + ̸=− = = ∈ ℕ ai k aj m, i j, i, j 1, 2, 3, k, m 0, (5.6) and let either one of the conditions in (2.12) or (2.14) hold. Let L L be the contour which separates all + ̸= + ̸= = −∞∈ ℕ poles cl in (2.1) to the left and all poles ai k, i 1, 2, 3, k 0, to the right. Then, the generalized hypergeometric function 3F2 a1, a2, a3; b1, b2; z dened by the Mellin–Barnes= integral (1.7) with L L−∞ has the power series representation (1.4). + = ∈ ℕ [ ] = Theorem 5.7. Let a1, a2, a3, b1, b2 be such that the conditions in (5.5)–(5.6) are satised and let either one of the conditions in (2.14) or (2.19) hold. Let L+∞ be the contour which separates all poles cl in (2.1) ∈ ℂ A. A. Kilbas et al., The generalized hypergeometric function as the Meijer G-function Ë 13

to the left and all poles ai k, i 1, 2, 3, k 0 to the right. Then, for the generalized hypergeometric function 3F2 a1, a2, a3; b1, b2; z dened by the Mellin–Barnes integral (1.7) with L L+∞ holds the formula + = ∈ ℕ a Γ b1 Γ b2 Γ a2 a1 Γ a3 a1 1 1 3F2[ a1, a2, a3; b1, b2; z] = Γ a2 Γ a3 Γ b1 a1 Γ b2 a1 z ( ) ( ) ( − ) ( − ) 1 ( ) = 3F2 a1, 1 a1 b1, 1 −a1 b2×; 1 a1 a2, 1 a1 a3; ( ) ( ) ( − ) ( − ) z a Γ ×b1 Γ b2 Γ a+1 a−2 Γ a3 + a2 − 1 2+ − + − Γ a1 Γ a3 Γ b1 a2 Γ b2 a2 z ( ) ( ) ( − ) ( − ) 1 + 3F2 a2, 1 a2 b1, 1 a2 −b2; 1 ×a2 a1, 1 a2 a3; ( ) ( ) ( − ) ( − ) z a Γ ×b1 Γ b2 Γ a+1 a−3 Γ a2 + a3 − 1 3+ − + − Γ a1 Γ a2 Γ b1 a3 Γ b2 a3 z ( ) ( ) ( − ) ( − ) 1 + 3F2 a3, 1 a3 b1, 1 a3 −b2; 1 ×a3 a1, 1 a3 a2; . ( ) ( ) ( − ) ( − ) z

Corollary 5.8. Let a1, a2, a3, b1, b2 ×be such that+ the conditions− + in−(5.5)–(5.6)+ are− satised.+ − Then, the generalized hypergeometric function 3F2 a1, a2, a3; b1, b2; z has the asymptotic estimate ∈ ℂ a Γ b1 Γ b2 Γ a2 a1 Γ a3 a1 1 1 1 3F2 a1, a2, a3; b1, b2; z [ ] 1 O Γ a2 Γ a3 Γ b1 a1 Γ b2 a1 z z ( ) ( ) ( − ) ( − ) [ ] = Γ b Γ b Γ a a Γ a − a 1+ a2 1 ( ) ( )1 ( 2− )1 ( 2− )3 2 1 O Γ a1 Γ a3 Γ b1 a2 Γ b2 a2 z z ( ) ( ) ( − ) ( − ) + Γ b Γ b Γ a a Γ a a −1a3 + 1 ( 1) ( 2) ( 1 − 3) ( 2 − 3) 1 O Γ a1 Γ a2 Γ b1 a3 Γ b2 a3 z z ( ) ( ) ( − ) ( − ) as z . In particular, + − + ( ) ( ) ( − ) 1( a∗− ) 3F2 a1, a2, a3; b1, b2; z O → +∞ z as z , where [ ] = ∗ a min Re a1 , Re a2 , Re a3 . → +∞ Finally, taking p q 3 in Theorem 4.1 and Theorem 4.3, we obtain the following result for the generalized = ( ) ( ) ( ) hypergeometric function 3F3 a1, a2a3; b1, b2, b3; z . = = Theorem 5.9. Let a , b , i, j 1, 2, 3, be such that i j [ ]

∈bℂj l,= ai k l, i 1, 2, 3, j 1, 2, 3, k, l 0, and ̸=− + ̸=− = = ∈ ℕ ai k aj m, i j, i, j 1, 2, 3, k, m 0, and let z 0. Let L be the contour which separates all poles c in (2.1) to the left and all poles a k, −∞ + ̸= + ̸= = l ∈ ℕ i i 1, 2, 3, k 0, to the right. Then, for the hypergeometric function 3F3 a1, a2, a3; b1, b2, b3; z dened by the Mellin–Barnes̸= integral + = ∈ ℕ [ ] 1 Γ s Γ a1 s Γ a2 s Γ a3 s −s 3F3 a1, a2, a3; b1, b2, b3; z z ds, 2πi Γ b1 s Γ b2 s Γ b3 s L−∞ ( ) ( − ) ( − ) ( − ) [ ] = X (− ) the series representation ( − ) ( − ) ( − )

+∞ a a a zk F a , a .a ; b , b , b ; z 1 k 2 k 3 k 3 3 1 2 3 1 2 3 b b b k! k=0 1 k 2 k 3 k ( ) ( ) ( ) [ ] = H holds. ( ) ( ) ( ) 14 Ë A. A. Kilbas et al., The generalized hypergeometric function as the Meijer G-function

Note: This is one of the last papers by Professor Anatoly Kilbas, who passed away on June 28, 2010.

Funding: The present research was supported by the Belarusian Fundamental Research Fund (project F03MC-008), by the Universidad de La Laguna, by the Science Promotion Fund of the Japan Private School Promotion Foundation and by a grant of the University Grants Commission of India.

References

[1] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, 1953. [2] A. A. Kilbas and M. Saigo, H-Transforms. Theory and Applications, Anal. Methods Spec. Funct. 9, Chapman & Hall / CRC, Boca Raton, 2004. [3] Y. Luke, The Special Functions and Their Approximations, Math. Sci. Eng. 53, Academic Press, New York, 1962. [4] A. M. Mathai, A Handbook of Generalized Special Functions for Statistical and Physical Sciences, The Clarendon Press, Oxford, 1993. [5] A. M. Mathai and R. K. Saxena, The H-function with Applications in Statistics and Other Disciplines, Halsted Press, New York, 1978. [6] A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and Series, Vol. 3, More Special Functions, Gordon and Breach, New York, 1990. [7] R. K. Saxena and S. L. Kalla, Asymptotic formulas for unied elliptic-type integrals, Integral Transforms Spec. Funct. 15 (2004), no. 4, 359–368. [8] R. K. Saxena, S. L. Kalla and J. H. Hubbell, Asymptotic expansion of a unied elliptic-type integral, Math. Balkanica (N.S.) 14 (2000), no. 3–4, 387–396. [9] R. K. Saxena and M. A. Pathan, Asymptotic formulas for unied elliptic-type integrals, Demonstratio Math. 36 (2003), no. 3, 579–589. [10] H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Function of One and Two Variables with Applications, South Asian Publishers, New Delhi, 1982.