• Kyungpook Mathematical Journal
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• v.57 no.3
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• pp.393-400
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• 2017

Very recently, Srivastava et al. [8] introduced an extension of the Pochhammer symbol and used it to define a generalization of the generalized hypergeometric functions. In this paper, by using the generalized Pochhammer symbol, we extend the generalized Hurwitz-Lerch Zeta function by Goyal and Laddha [6] and investigate some interesting properties which include various integral representations, Mellin transforms, differential formula and generating function. Some interesting special cases of our main results are also considered.

• Honam Mathematical Journal
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• v.22 no.1
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• pp.53-60
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• 2000

We evaluate some interesting families of infinite series expressed in terms of the Psi (or Digamma) and Zeta functions by analyzing the well-known identity associated with $_3F_2$ due to Watson. Some special cases are also considered.

• Communications of the Korean Mathematical Society
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• v.11 no.1
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• pp.57-62
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• 1996

For $1 < p \leq 2$ it is shown that a certain sequence of functions converges to -1 in $L^{p-\varepsilon}(0, 1)$ for any small $\varepsilon > 0$ if and only if the Riemann zeta function satisfies $\zeta(s) \neq 0$ for $\sigma = Re s > 1/p$.

• Journal of the Korean Mathematical Society
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• v.44 no.5
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• pp.1163-1184
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• 2007

In this presentation dedicated to the tricentennial birth anniversary of the great eighteenth-century Swiss mathematician, Leonhard Euler (1707-1783), we begin by remarking about the so-called Basler problem of evaluating the Zeta function ${\zeta}(s)$ [in the much later notation of Georg Friedrich Bernhard Riemann (1826-1866)] when s=2, which was then of vital importance to Euler and to many other contemporary mathematicians including especially the Bernoulli brothers [Jakob Bernoulli (1654-1705) and Johann Bernoulli (1667-1748)], and for which a fascinatingly large number of seemingly independent solutions have appeared in the mathematical literature ever since Euler first solved this problem in the year 1736. We then investigate various recent developments on the evaluations and representations of ${\zeta}(s)$ when $s{\in}{\mathbb{N}}{\backslash}\;[1],\;{\mathbb{N}}$ being the set of natural numbers. We emphasize upon several interesting classes of rapidly convergent series representations for ${\zeta}(2n+1)(n{\in}{\mathbb{N}})$ which have been developed in recent years. In two of many computationally useful special cases considered here, it is observed that ${\zeta}(3)$ can be represented by means of series which converge much more rapidly than that in Euler's celebrated formula as well as the series used recently by Roger $Ap\'{e}ry$ (1916-1994) in his proof of the irrationality of ${\zeta}(3)$. Symbolic and numerical computations using Mathematica (Version 4.0) for Linux show, among other things, that only 50 terms of one of these series are capable of producing an accuracy of seven decimal places.

• Honam Mathematical Journal
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• v.39 no.4
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• pp.561-568
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• 2017

Hurwitz zeta function occurs in various parts of mathematics. In particular, it plays an important role in some area of number theory. In this paper, using a certain transformation formula, we find some identities of relations between ${\zeta}(2n+1)$ and ${\zeta}(2n+1,{\alpha})$ for special values of ${\alpha}$.

• East Asian mathematical journal
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• v.18 no.2
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• pp.219-224
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• 2002

Recently the theory of the multiple Gamma functions, which were studied by Barnes and others a century ago, has been revived in the study of determinants of Laplacians. Here we are aiming at evaluating the values of the multiple Gamma functions ${\Gamma}_n(\frac{1}{2})$ in terms of the Hurwitz or Riemann Zeta functions.

• Journal of applied mathematics & informatics
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• v.41 no.6
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• pp.1409-1418
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• 2023

The Hurwitz-type Euler zeta function ζ_E(z, q) is defined by the series ${\zeta}_E(z,\,q)\,=\,\sum\limits_{n=0}^{\infty}{\frac{(-1)^n}{(n\,+\,q)^z}},$ for Re(z) > 0 and q ≠ 0, -1, -2, . . . , and it can be analytic continued to the whole complex plane. An asymptotic expansion for ζ^'_E(-m, q) has been proved based on the calculation of Hermite's integral representation for ζ_E(z, q).

• Journal of the Korean Mathematical Society
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• v.57 no.6
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• pp.1535-1549
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• 2020

This paper deals with the inverse of tails of Hurwitz zeta function. More precisely, for any positive integer s ≥ 2 and 0 ≤ a < 1, we give an algorithm for finding a simple form of f_s,a(n) such that $$\lim_{n{\rightarrow}{\infty}}\{\({\sum\limits_{k=n}^{\infty}}{\frac{1}{(k+a)^s}}\)^{-1}-f_{s,a}(n)\}=0$$. We show that f_s,a(n) is a polynomial in n-a of order s-1. All coefficients of f_s,a(n) are represented in terms of Bernoulli numbers.

• East Asian mathematical journal
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• v.31 no.1
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• pp.41-53
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• 2015

A large number of series and integral representations for the Stieltjes constants (or generalized Euler-Mascheroni constants) ${\gamma}_k$ and the generalized Stieltjes constants ${\gamma}_k(a)$ have been investigated. Here we aim at presenting certain integral representations for the generalized Stieltjes constants ${\gamma}_k(a)$ by choosing to use four known integral representations for the generalized zeta function ${\zeta}(s,a)$. As a by-product, our main results are easily seen to specialize to yield those corresponding integral representations for the Stieltjes constants ${\gamma}_k$. Some relevant connections of certain special cases of our results presented here with those in earlier works are also pointed out.

• Journal of the Korean Mathematical Society
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• v.43 no.6
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• pp.1269-1287
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• 2006

As a continuation of computing the zeta function of a regular covering graph by Mizuno and Sato in [9], we derive in this paper computational formulae for the zeta functions of a graph bundle and of any (regular or irregular) covering of a graph. If the voltages to derive them lie in an abelian or dihedral group and its fibre is a regular graph, those formulae can be simplified. As a by-product, the zeta function of the cartesian product of a graph and a regular graph is obtained. The same work is also done for a discrete torus and for a discrete Klein bottle.