S S symmetry Article Note on the Hurwitz–Lerch Zeta Function of Two Variables Junesang Choi 1, Recep ¸Sahin 2, O˘guzYa˘gcı 2 and Dojin Kim 3,* 1 Department of Mathematics, Dongguk University, Gyeongju 38066, Korea; [email protected] 2 Department of Mathematics, Kırıkkale University, Kırıkkale 71450, Turkey; [email protected] (R.S.); [email protected] (O.Y.) 3 Department of Mathematics, Dongguk University, Seoul 04620, Korea * Correspondence: [email protected] Received: 3 August 2020; Accepted: 27 August 2020; Published: 28 August 2020 Abstract: A number of generalized Hurwitz–Lerch zeta functions have been presented and investigated. In this study, by choosing a known extended Hurwitz–Lerch zeta function of two variables, which has been very recently presented, in a systematic way, we propose to establish certain formulas and representations for this extended Hurwitz–Lerch zeta function such as integral representations, generating functions, derivative formulas and recurrence relations. We also point out that the results presented here can be reduced to yield corresponding results for several less generalized Hurwitz–Lerch zeta functions than the extended Hurwitz–Lerch zeta function considered here. For further investigation, among possibly various more generalized Hurwitz–Lerch zeta functions than the one considered here, two more generalized settings are provided. Keywords: beta function; gamma function; Pochhammer symbol; Hurwitz–Lerch zeta function; Hurwitz–Lerch zeta function of two variables; hypergeometric functions; confluent hypergeometric functions; Appell hypergeometric functions; Humbert hypergeometric functions of two variables; integral representations; generating functions; derivative formulas; recurrence relation 1. Introduction and Preliminaries The generalized (or Hurwitz) zeta function z(s, n) is defined by (see, e.g., [1], pp. 24–27); see also ([2] Chapter XIII) ¥ 1 z(s, n) = <(s) > 1, n 2 n − , (1) ∑ ( + )s C Z0 n=0 n n which is a generalization of the Riemann zeta function z(s) := z(s, 1) (see, e.g., [1] Section 1.12). Apostol [3] showed that the following analytic continuation formula (see, e.g., ([1] p. 26, Equation (6)), ([4] Equation (5.1)) 2 G(s) ¥ cos(ps/2 − 2pnn) z(1 − s, n) = (0 < n ≤ 1, <(s) > 1), (2) ( )s ∑ s 2p n=1 n where G(s) is the familiar Gamma function whose Euler’s integral (see, e.g., [1] pp. 1–24) is ¥ Z G(s) = e−u us−1 du (<(s) > 0), (3) 0 Symmetry 2020, 12, 1431; doi:10.3390/sym12091431 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 1431 2 of 11 and it can be derived from a known transformation formula for the Lerch zeta function ¥ e2pinx f(x, n, s) = <(s) > 1, n 2 n −, x 2 . (4) ∑ ( + )s C Z0 R n=0 n n Note that It is easy to read that f(x, n, s) = z(s, n) when x 2 Z. The Hurwitz–Lerch zeta function F (z, s, n) is defined by (see, e.g., [1] p. 27) ¥ zn F (z, s, n) = ∑ s , (5) n=0 (n + n) − where n 2 C n Z0 . The F (z, s, n) in (5) converges for all s 2 C when jzj < 1 and for <(s) > 1 when jzj = 1. Here and elsewhere, let N, Z, R, and C be the sets of positive integers, integers, real numbers, − and complex numbers, respectively. Furthermore, let us denote Z0 := Z n N and N0 := N [ f0g. The various special cases of the Hurwitz–Lerch zeta function (5) including the Riemann zeta function z(s), the Hurwitz zeta function (1), and the Lerch zeta function (4) have been intensively studied and applied. We choose to take some examples: Adamchik and Srivastava ([5] Propostion 5) evaluated a series involving polygamma functions in terms of the Hurwitz–Lerch zeta function (5). Rassias and Yang [6] studied certain equivalent conditions of a reverse Hilbert-type integral inequality, for which, in an example, the generalized zeta function z(s, a) is shown to be related to a best possible constant factor (see also [7? ,8]). Recently, a number of generalizations of the Hurwitz–Lerch zeta function (5) have been actively investigated (see, e.g., [9–18] and the references cited therein). Furthermore, very recently, Choi and Parmar [19] have introduced and investigated the following two-variable extension of the Hurwitz–Lerch zeta function (5) ¥ 0 k ` (a)k+` (b)k (b )` x y F 0 (x, y, s, a) = , (6) a,b,b ;c ∑ ( ) ` ( + ` + )s k,`=0 c k+` k! ! k a 0 − where a, b, b 2 C and c, a 2 C n Z0 . The function Fa,b,b0;c (x, y, s, a) in (6) converges for all s 2 C 0 when jxj < 1 and jyj < 1, and for <(s + c − a − b − b ) > 1 when jxj = 1 and jyj = 1. Here (h)n is the Pochhammer symbol given (for h, n 2 C) by G(h + n) − (h)n : = (h + n 2 C n Z ) G(h) 0 ( (7) 1 (n = 0), = h(h + 1) ··· (h + n − 1)(n = n 2 N). From (3) and (7), the following integral formula for the Pochhammer symbol ¥ 1 Z (s) = e−u us+n−1 du (<(s + n) > 0) (8) n G(s) 0 can be easily obtained. Here, in a systematic way, we aim to establish certain formulas and representations for the extended Hurwitz–Lerch zeta function of two variables (6) such as integral representations, generating functions, derivative formulas and recurrence relations. We also point out that the results presented here can be reduced to produce corresponding results for several less generalized Hurwitz–Lerch zeta functions than the extended Hurwitz–Lerch zeta function (6). Further, two more generalized settings than (6) are provided. Symmetry 2020, 12, 1431 3 of 11 2. Integral Representations for the Extended Hurwitz–Lerch Zeta Function of Two Variables We begin by recalling a known integral representation of the extended Hurwitz–Lerch zeta function (6) (see [19] Theorem 1) ¥ Z 1 s−1 −au 0 −u −u F 0 (x, y, s, a) = u e F a, b, b ; c; xe , ye du, (9) a,b,b ;c G(s) 1 0 which converges for min f<(s), <(a)g > 0 when jxj ≤ 1 (x 6= 1) and jyj ≤ 1 (y 6= 1), and for <(s) > 1 when x = 1 and y = 1. Here F1 is the Appell hypergeometric function of two variables defined by (see, e.g., ([1] p. 224, Equation (6)); see also ([20] p. 22)) ¥ 0 m n (a) + (b) (b ) x y F a, b, b0; c; x, y = m n m n 1 ∑ ( ) m,n=0 c m+n m! n! " 0 # (10) ¥ (a) (b) a + m, b ; xm = m m F y , ∑ ( ) 2 1 m=0 c m c + m ; m! 0 − where a, b, b 2 C, c 2 C n Z0 and whose convergence region is maxf<(x), <(y)g < 1. Here pFq (p, q 2 N0) are the generalized hypergeometric functions (see, e.g., ([1] Chapters II and V); see also ([21] Section 1.5), [20,22–25]). The following confluent form of the Appell hypergeometric function F1 is recalled (see, e.g., [1] p. 225, Equation (21)); see also ([20] p. 22 et seq.) ¥ (b) (b0) xm yn F [b, b0; c; x, y] = m n (jxj < ¥, jyj < ¥). (11) 2 ∑ ( ) m,n=0 c m+n m! n! We provide further integral representations of the extended Hurwitz–Lerch zeta function (6), asserted in the following theorem. Theorem 1. Each of the following integral representations holds. ¥ ¥ ¥ Z Z Z 1 s−1 b−1 b0−1 F 0 (x, y, s, a) = u v v exp(−au − v − v ) a,b,b ;c G(s) G(b)G(b0) 1 2 1 2 0 0 0 (12) −u × 1F1 a; c; e (xv1 + yv2) du dv1 dv2, where min f<(s), <(a), <(b), <(b0)g > 0; ¥ ¥ ¥ ¥ Z Z Z Z 1 s−1 b−1 b0−1 a−1 F 0 (x, y, s, a) = u v v v a,b,b ;c G(s) G(a)G(b)G(b0) 1 2 3 0 0 0 0 (13) −u × exp(−au − v1 − v2 − v3)0F1 −; c; e (xv1 + yv2) v3 du dv1 dv2 dv3, where min f<(s), <(a), <(a), <(b), <(b0)g > 0; ¥ ¥ Z Z 1 s−1 a−1 F 0 (x, y, s, a) = u v exp(−au − v) a,b,b ;c G(s) G(a) 0 0 (14) 0 −u −u × F2 b, b ; c; xe v, ye v du dv, Symmetry 2020, 12, 1431 4 of 11 where minf<(s), <(a), <(a)g > 0 and maxf<(x), <(y)g < 1; G(c) F 0 (x, y, s, a) = a,b,b ;c G(s) G(a) G(b)G(b0)G(c − b − b0) ¥ ¥ 1 1 Z Z Z Z 0 × us−1 va−1 xb−1 hb −1 (1 − x)c−b−1 (15) 0 0 0 0 0 × (1 − h)c−b−b −1 exp −au − v + xe−uvx + ye−uv(1 − x)h dx dh du dv, where minf<(s), <(a), <(a), <(b)g > 0 and <(c − b) > <(b0) > 0; ¥ ¥ 1 1 Z Z Z Z G (d) s−1 a−1 b−1 b0−1 F 0 (x, y, s, a) = u v x h a,b,b ;c G (s) G (a) G (b) G (b0) G (d − b − b0) 0 0 0 0 − − − − 0− (16) × (1 − x)d b 1 (1 − h)d b b 1 exp −au − v + xe−uvx + ye−uv (1 − x) h −u −u × 1F1 c − d; c; −ze vx − ye v (1 − x) h dx dh du dv, where minf<(s), <(a), <(a), <(b)g > 0 and <(d − b) > <(b0) > 0; G (c) F 0 (x, y, s, a) = a,b,b ;c G (s) G (a) G (d) G (b0) G (c − d − b0) ¥ ¥ 1 1 Z Z Z Z 0 0 × us−1va−1xd−1hb −1 (1 − x)c−d−1 (1 − h)c−d−b −1 (17) 0 0 0 0 −u −u −u × exp −au − v + xe vx + ye v(1 − x)h 1F1 d − b; d; −xe vx dx dh du dv, where minf<(s), <(a), <(a), <(d)g > 0 and <(c − d) > <(b0) > 0; ¥ ¥ 1 1 Z Z Z Z G (c) s−1 a−1 d−1 b0−1 F 0 (x, y, s, a) = u v x h a,b,b ;c G (s) G (a) G (d) G (b0) G (c − d − b0) 0 0 0 0 0 (18) × (1 − x)c−d−1(1 − h)c−d−b −1 exp −au − v + xe−uvx + ye−uv (1 − x) h 0 −u × 1F1 b − d; c − d − b ; xe v (1 − x)(1 − h) dx dh du dv, where minf<(s), <(a), <(a), <(d)g > 0 and <(c − d) > <(b0) > 0; ¥ ¥ 1 1 G (c) Z Z Z Z s−1 a−1 d1−1 d2−1 Fa,b,b0;c (x, y, s, a) = u v x h G (s) G (a) G (d1) G (d2) G (c − d1 − d2) 0 0 0 0 (19) c− − c− − − −u −u × (1 − x) d1 1(1 − h) d1 d2 1 exp −au − v + xe vx + ye v(1 − x)h −u 0 −u × 1F1 d1 − b; d1; −xe vx 1F1 d2 − b ; d2; −ye v(1 − x)h dx dh du dv, where minf<(s), <(a), <(a), <(d1)g > 0 and <(c − d1) > <(d2) > 0; G (c) Fa,b,b0;c (x, y, s, a) = G (s) G (a) G (d1) G (d2) G (c − d1 − d2) ¥ ¥ 1 1 Z Z Z Z − − − − − × us−1va−1xd1−1hd2−1 (1 − x)c d1 1 (1 − h)c d1 d2 1 (20) 0 0 0 0 × exp −au − v + xe−uvx + ye−uv (1 − x) h 0 −u −u × F2 b − d1; b − d2; c; xe v(1 − x)(1 − h), ye v(1 − x)(1 − h)] dx dh du dv, Symmetry 2020, 12, 1431 5 of 11 where minf<(s), <(a), <(a), <(d1)g > 0 and <(c − d1) > <(d2) > 0; G (c) F 0 (x, y, s, a) = a,b,b ;c G (s) G (a) G (b) G (b0) G (c − b − b0) ¥ 1 1−h Z Z Z 0 − − 0− × e−au us−1xb−1hb −1 (1 − x − h)c b b 1 (21) 0 0 0 −a × 1 − xxe−u − yhe−u dx dh du, where minf<(s), <(a), <(b), <(b0)g > 0 and <(c − b − b0) > 0; G (c) F 0 (x, y, s, a) = a,b,b ;c G (s) G (a) G (c − a) ¥ 1 (22) Z Z − − −b −b0 × e−au us−1va−1 (1 − v)c a 1 1 − vxe−u 1 − vye−u dv du, 0 0 where minf<(s), <(a), <(a)g > 0 and <(c − a) > 0; ¥ 1 Z Z G (c) −au s−1 c−b−1 b−1 F 0 (x, y, s, a) = e u v (1 − v) a,b,b ;c G (s) G (b) G (c − b) 0 0 (23) −uc−a−b −ua−c 0 −u × 1 − xe 1 − vxe 2F1 a, b ; c − b; ye v dv du, where minf<(s), <(a), <(b)g > 0 and <(c − b) > 0.