• Title/Summary/Keyword: Hurwitz's Zeta Function

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SPECIAL VALUES AND INTEGRAL REPRESENTATIONS FOR THE HURWITZ-TYPE EULER ZETA FUNCTIONS

  • Hu, Su;Kim, Daeyeoul;Kim, Min-Soo
    • Journal of the Korean Mathematical Society
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    • v.55 no.1
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    • pp.185-210
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    • 2018
  • The Hurwitz-type Euler zeta function is defined as a deformation of the Hurwitz zeta function: $${\zeta}_E(s,x)={\sum_{n=0}^{\infty}}{\frac{(-1)^n}{(n+x)^s}}$$. In this paper, by using the method of Fourier expansions, we shall evaluate several integrals with integrands involving Hurwitz-type Euler zeta functions ${\zeta}_E(s,x)$. Furthermore, the relations between the values of a class of the Hurwitz-type (or Lerch-type) Euler zeta functions at rational arguments have also been given.

AN ASYMPTOTIC EXPANSION FOR THE FIRST DERIVATIVE OF THE HURWITZ-TYPE EULER ZETA FUNCTION

  • MIN-SOO KIM
    • 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 ζE(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 ζE(z, q).

ASYMPTOTIC BEHAVIOR OF THE INVERSE OF TAILS OF HURWITZ ZETA FUNCTION

  • Lee, Ho-Hyeong;Park, Jong-Do
    • 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 fs,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 fs,a(n) is a polynomial in n-a of order s-1. All coefficients of fs,a(n) are represented in terms of Bernoulli numbers.

MULTIPLICATION FORMULA AND (w, q)-ALTERNATING POWER SUMS OF TWISTED q-EULER POLYNOMIALS OF THE SECOND KIND

  • CHOI, JI EUN;KIM, AHYUN
    • Journal of applied mathematics & informatics
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    • v.39 no.3_4
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    • pp.455-467
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    • 2021
  • In this paper, we define twisted q-Euler polynomials of the second kind and explore some properties. We find generating function of twisted q-Euler polynomials of the second kind. Also, we investigate twisted q-Raabe's multiplication formula and (w, q)-alternating power sums of twisted q-Euler polynomials of the second kind. At the end, we define twisted q-Hurwitz's type Euler zeta function of the second kind.

NEW CLASS OF INTEGRALS INVOLVING GENERALIZED HYPERGEOMETRIC FUNCTION AND THE LOGARITHMIC FUNCTION

  • Kim, Yongsup
    • Communications of the Korean Mathematical Society
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    • v.31 no.2
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    • pp.329-342
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    • 2016
  • Motivated essentially by Brychkov's work [1], we evaluate some new integrals involving hypergeometric function and the logarithmic function (including those obtained by Brychkov[1], Choi and Rathie [3]), which are expressed explicitly in terms of Gamma, Psi and Hurwitz zeta functions suitable for numerical computations.

INFINITE SERIES ASSOCIATED WITH PSI AND ZETA FUNCTIONS

  • KIM, YONGSUP
    • 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.

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NOTE ON CAHEN′S INTEGRAL FORMULAS

  • Choi, June-Sang
    • Communications of the Korean Mathematical Society
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    • v.17 no.1
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    • pp.15-20
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    • 2002
  • We present an explicit form for a class of definite integrals whose special cases include some definite integrals evaluated, over a century ago, by Cahen who made use of an appropriate contour integral for the integrand of a well-known integral representation of the Riemann Zeta function given in (3). Furthermore another analogous class of definite integral formulas and some identities involving Riemann Zeta function and Euler numbers En are also obtained as by-products.

DETERMINANTS OF THE LAPLACIANS ON THE n-DIMENSIONAL UNIT SPHERE Sn (n = 8, 9)

  • Choi, June-Sang
    • Honam Mathematical Journal
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    • v.33 no.3
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    • pp.321-333
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    • 2011
  • During the last three decades, the problem of evaluation of the determinants of the Laplacians on Riemann manifolds has received considerable attention by many authors. The functional determinant for the n-dimensional sphere $S^n$ with the standard metric has been computed in several ways. Here we aim at computing the determinants of the Laplacians on $S^n$ (n = 8, 9) by mainly using ceratin known closed-form evaluations of series involving Zeta function.

EULER SUMS EVALUATABLE FROM INTEGRALS

  • Jung, Myung-Ho;Cho, Young-Joon;Choi, June-Sang
    • Communications of the Korean Mathematical Society
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    • v.19 no.3
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    • pp.545-555
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    • 2004
  • Ever since the time of Euler, the so-called Euler sums have been evaluated in many different ways. We give here a proof of the classical Euler sum by following Lewin's method. We also consider some related formulas involving Euler sums, which are evaluatable from some known definite integrals.