By H. M. Srivastava

Zeta and q-Zeta capabilities and linked sequence and Integrals is a completely revised, enlarged and up-to-date model of sequence linked to the Zeta and comparable services. a number of the chapters and sections of the ebook were considerably converted or rewritten and a brand new bankruptcy at the idea and purposes of the elemental (or q-) extensions of assorted specific services is incorporated. This publication may be useful because it covers not just distinct and systematic shows of the idea and functions of a few of the equipment and strategies utilized in facing many alternative periods of sequence and integrals linked to the Zeta and similar features yet stimulating old debts of a big variety of difficulties and well-classified tables of sequence and integrals.Detailed and systematic displays of the idea and purposes of a few of the tools and strategies utilized in facing many various periods of sequence and integrals linked to the Zeta and comparable capabilities

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12 2 (n − 1)2 (73) Now apply the following special case of (53) when n = z = 1: ψ (1) = ζ (2) = π2 , 6 (74) in conjunction with (68), and (73) readily yields n ψ (k) = n k=1 =n π2 1 1 1 1 2 n−1 − 2 + 2 + ··· + + 2 + 2 + ··· + 6 1 2 (n − 1)2 1 2 (n − 1)2 1 1 + + · · · + γ + log(n − 1) + O n−1 n2 (n + 1)2 (n → ∞), (75) which, by applying (70), produces the desired result (67). 4 The Multiple Gamma Functions The double Gamma function 2 and the multiple Gamma functions n were defined and studied systematically by Barns [94–97] in about 1900.

12 2 (n − 1)2 (73) Now apply the following special case of (53) when n = z = 1: ψ (1) = ζ (2) = π2 , 6 (74) in conjunction with (68), and (73) readily yields n ψ (k) = n k=1 =n π2 1 1 1 1 2 n−1 − 2 + 2 + ··· + + 2 + 2 + ··· + 6 1 2 (n − 1)2 1 2 (n − 1)2 1 1 + + · · · + γ + log(n − 1) + O n−1 n2 (n + 1)2 (n → ∞), (75) which, by applying (70), produces the desired result (67). 4 The Multiple Gamma Functions The double Gamma function 2 and the multiple Gamma functions n were defined and studied systematically by Barns [94–97] in about 1900.

2! · · · n! )n 1 · 2 · 32 · 43 · · · nn−1 (n ∈ N). (9) We shall deduce only the expression (4) (see Barnes [94]). Indeed, taking the logarithmic derivative on both sides of the fundamental functional relation in (6), with respect to z, we obtain G (z + 1) = G(z + 1) (z) G (z) + , (z) G(z) from which G (z + n + 2) = G(z + n + 2) n k=0 (z + k + 1) G (z + 1) + . (z + k + 1) G(z + 1) For sufficiently small values of |z|, by Taylor’s theorem, we have (z + 1 + k) = (z + 1 + k) (1 + k) d +z (1 + k) dk (1 + k) z2 d2 + (1 + k) 2!

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