ON THE H-FUNCTION

The paper is devoted to study the $H$-function defined by the Mellin-Barnes integral $$H^{m,n}_{\thinspace p,q}(z)={\frac1{2\pi i}}\int_{\Lss} \HHs^{m,n}_{\thinspace p,q}(s)z^{-s}ds,$$ where the function $\HH^{m,n}_{\thinspace p,q}(s)$ is a certain ratio of products of Gamma functions with the argument $s$ and the contour $\LL$ is specially chosen. The conditions for the existence of $H^{m,n}_{\thinspace p,q}(z)$ are discussed and explicit power and power-logarithmic series expansions of $H^{m,n}_{p,q}(z)$ near zero and infinity are given. The obtained results define more precisely the known results.


Introduction
This paper deals with the H-function H m,n p,q (z) introduced by Pincherle in 1888 (see [3,Section 1.19]). Interest in this function appeared in 1961, when Fox [4] investigated such a function as symmetrical Fourier kernel. Therefore, the H-function is often called Fox's H-function. For integers m, n, p, q such that 0 ≦ m ≦ q, 0 ≦ n ≦ p and a i , b j ∈ C and α i , β j ∈ R + = (0, ∞) (1 ≦ i ≦ p, 1 ≦ j ≦ q) the function is defined by the Mellin , (1.2) the contour L is specially chosen and an empty product, if it occurs, is taken to be one. The theory of this function may be found in [10], [2], [1], [9,Chapter 2], [12,Chapter 1] and [11,Section 8.3]. We only indicate that most of the elementary and special functions are particular cases of the H-function H m,n p,q (z). In particular, if α's and β's are equal to 1, the H-function (1.1) reduces to Meijer's G-function G m,n p,q (z). The conditions of the existence of the H-function can be made by inspecting the convergence of the integral (1.1), which depend on the selection of the contour L and the relations between parameters a i , α i (i = 1, · · · , p) and b j , β j (j = 1, · · · , q). Especially, the relations may depend on the numbers ∆, δ and µ defined by Such a selection of L and the relations on parameters are indicated in the handbook [11,Section 8.3.1], but some of the results there needs correction. In this paper we would like to give such a correction in the following cases: (a) ∆ ≧ 0 and the contour L = L −∞ in (1.1) runs from −∞ + iϕ 1 to −∞ + iϕ 2 , ϕ 1 < ϕ 2 , such that the poles of Γ(b j + β j s) (j = 1, · · · , m) lie on the left of L −∞ and those of Γ(1 − a i − α i s) (i = 1, · · · , n) on the right of L −∞ .
Our results are based on the asymptotic behavior of the function H m,n p,q (s) given in (1.2) at infinity. Using the behavior and following [1], we give the series representation of H m,n p,q (z) via residues of the integrand H m,n p,q (s)z −s . In this way we simplify the proof of Theorem 1 in [1] by applying the former results to find the explicit series expansions of H m,n p,q (z). Such power expansions, as corollaries of the results from [1], were indicated in [12, Chapter 2.2] (see also [11,Section 8.3.1]), provided that the poles of Gamma-functions Γ(b j + β j s) (j = 1, · · · , m) and Γ(1 − a i − α i s) (i = 1, · · · , n) do not coincide β j (a i − 1 − k) = α i (b j + l) (i = 1, · · · , n; j = 1, · · · , m; k, l ∈ N 0 = {0, 1, 2, · · ·}) (1. 6) in the cases: (c) ∆ > 0 with z = 0 or ∆ = 0 with 0 < |z| < δ, and the poles of Gamma-functions Γ(b j + β j s) (j = 1, · · · , m) are simple: (d) ∆ < 0 with z = 0 or ∆ = 0 with |z| > δ, and the poles of Gamma-functions Γ(1 − a i − α i s) (i = 1, · · · , n) are simple: When the poles of Gamma-functions in (c) and (d) coincide, explicit series expansions of H m,n p,q (z) should be more complicated power-logarithmic expansions. Such expansions in particular cases of the Meijer's G-functions G p,0 0,p and G p,0 p,p and of the H-functions H p,0 0,p and H p,0 p,p were given in [7] and [8], respectively. We obtain the explicit expansions of the H-function of general form H m,n p,q (z) under the conditions in (1.6). We show that, if the poles of the Gamma-functions Γ(b j + β j s) (j = 1, · · · , m) and Γ(1 − a i − α i s) (i = 1, · · · , n) coincide in the cases (c) and (d), respectively, then the H-function (1.1) has power-logarithmic series expansions. In particular, we give the asymptotic expansions of H m,n p,q (z) near zero. We note that the obtained results will be different in the cases when either ∆ ≧ 0 or ∆ ≦ 0.
The paper is organized as follows. Section 2 is devoted to the conditions of the existence of the H-function (1.1) which are based on the asymptotic behavior of H m,n p,q (s) at infinity. Here we also give the representations of (1.1) via the residues of the integrand. The latter result is applied in Sections 3 and 4 to obtain the explicit power and power-logarithmic series expansions of H m,n p,q (z) and, in particular, its asymptotic estimates near zero.
Remark 1. The relation [3, (1.18.6)] needs correction with addition of the multiplier e x in the left hand side and it must be replaced by Next assertion gives the asymptotic behavior of H m,n p,q (s) defined in (1.2) at infinity on lines parallel to the real axis.

Remark 2.
The asymptotic estimate of the function H m,n p,q (s) at infinity on lines parallel to the imaginary axis H m,n p,q (σ+it) as |t| → ∞ was given in our paper with Shlapakov [5].
By appealing to Lemma 2, we give conditions of the existence of the H-function (1.1) with the contour L being chosen as indicated in (a) and (b) in Section 1.
Proof. Let us first consider the case (a) for which ∆ ≧ 0 and L = L −∞ . We have to investigate the convergence of the integral (1.1) on the lines as t → −∞. According to (2.10), we have the following asymptotic estimate for the integrand of (1.1): where B 1 and B 2 are given by (2.11) with σ being replaced by ϕ 1 and ϕ 2 , respectively. It follows from (2.21) that the integral (1.1) is convergent if and only if one of the conditions in (2.14) to (2.16) is satisfied.
In the case (b), ∆ ≦ 0 and the contour L is taken to be L +∞ . Then we have to investigate the convergence of the integral (1.1) on the lines l 1 and l 2 in (2.20), as t → +∞. By virtue of (2.8) and (2.9), we have the asymptotic estimate: where A 1 and A 2 are given by (2.9) with σ being replaced by ϕ 1 and ϕ 2 , respectively. Thus is given. But our proof of Theorem 2 along the ideas of [1] is more simple and is based on the asymptotic estimate of H m,n p,q (s) at infinity given in Lemma 2.

Explicit Power Series Expansions
In this section we apply Theorem 2 to obtain explicit power series expansions of the H-function (1.1) under the condition (1.6) in the case of (1.7) or (1.8).
First we consider the former case. By Theorem 2(A), we have to evaluate the residues of H(s)z −s at the points s = b jl given in (2.23), where and in what follows we simplify H m,n p,q (s) by H(s). To evaluate these residues we use the property of the Gamma-function [6, (3.30)], that is, in a neighbourhood of the poles z = −k (k ∈ N 0 ) the Gamma-function Γ(z) can be expanded in powers of z + k = ǫ Since the poles b jl are simple, i.e., the conditions in (1.7) hold,
Corollary 3. Let the conditions in (1.6) and (1.7) be satisfied, and let ∆ ≧ 0 and j 0 (1 ≦ j 0 ≦ m) be an integer such that Then there holds the asymptotic estimate: where h * j 0 is given by (3.6) with j = j 0 . In particular, Now we consider the case (1.8) when the poles of the Gamma-functions Γ(1−a i −α i s) (i = 1, · · · , n) are simple. By (3.1), evaluating the residues of H(s)z −s at the points a ik given in (2.24) we have similarly to the previous argument that Res s=a ik [H(s)z −s ] = −h ik z −a ik (i = 1, · · · , n; k ∈ N 0 ), (3.10) where a ik are given by (2.24) and Thus from Theorem 2(B) we have where the constants h ik are given by (3.11).
First we consider the case (e). Let b ≡ b jl be one of points (2.23) for which some poles of the Gamma-functions Γ(b j + β j s) (j = 1, · · · , m) coincide and N * ≡ N * jl be order of this pole. It means that there exist j 1 , · · · , j N * ∈ {1, · · · , m} and l j 1 , · · · , l j N * ∈ N 0 such that Then H(s)z −s has the pole of order N * at b and hence We denote .

(4.3)
Using the Leibniz rule, we have

Thus, in view of Theorem 2(A), we have
Theorem 5. Let the conditions in (1.6) be satisfied and let either ∆ > 0, z = 0 or ∆ = 0, 0 < |z| < δ. Then the H-function (1.1) has the power-logarithmic series expansion Here ′ and ′′ are summations taken over j, l (j = 1, · · · , m; l = 0, 1, · · ·) such that the Gamma-functions Γ(b j + β j s) have simple poles and poles of order N * jl at the points b jl , respectively, and the constants h * jl are given by (3.3) while the constants H * jli are given by (4.5).
Corollary 6. If the conditions in (1.6) are satisfied and ∆ ≧ 0, then (4.7) gives the asymptotic expansion of H m,n p,q (z) near zero and the main terms of this asymptotic formula have the form: Here ′ and ′′ are summations taken over j (j = 1, · · · , m) such that the Gamma-functions Γ(b j + β j s) have simple poles and poles of order N * j ≡ N * j0 at the points b j0 , respectively, and h * j are given by (3.6) while H * j are given by (4.6). and, then, find similarly to (4.4) and (4.5 In particular, if we set k = 0, j = N i0 − 1 and N i0 ≡ N i , then, using (4.15) and (3.1), we have Here ′ and ′′ are summations taken over i, k (i = 1, · · · , n; k = 0, 1, · · ·) such that Gamma-functions Γ(1 − a i − α i s) have simple poles and poles of order N ik at the points a ik , respectively, and the constants h ik are given by (3.11) while the constants H ikj are given by (4.17).