• Title/Summary/Keyword: Euler zeta function

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ON THE (p, q)-ANALOGUE OF EULER ZETA FUNCTION

  • RYOO, CHEON SEOUNG
    • Journal of applied mathematics & informatics
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    • v.35 no.3_4
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    • pp.303-311
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    • 2017
  • In this paper we define (p, q)-analogue of Euler zeta function. In order to define (p, q)-analogue of Euler zeta function, we introduce the (p, q)-analogue of Euler numbers and polynomials by generalizing the Euler numbers and polynomials, Carlitz's type q-Euler numbers and polynomials. We also give some interesting properties, explicit formulas, a connection with (p, q)-analogue of Euler numbers and polynomials. Finally, we investigate the zeros of the (p, q)-analogue of Euler polynomials by using computer.

SYMMETRIC IDENTITIES INVOLVING THE MODIFIED (p, q)-HURWITZ EULER ZETA FUNCTION

  • KIM, A HYUN;AN, CHAE KYEONG;LEE, HUI YOUNG
    • Journal of applied mathematics & informatics
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    • v.36 no.5_6
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    • pp.555-565
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    • 2018
  • The main subject of this paper is to introduce the (p, q)-Euler polynomials and obtain several interesting symmetric properties of the modified (p, q)-Hurwitz Euler Zeta function with regard to (p, q) Euler polynomials. In order to get symmetric properties, we introduce the new (p, q)-analogue of Euler polynomials $E_{n,p,q}(x)$ and numbers $E_{n,p,q}$.

A NOTE ON RECURRENCE FORMULA FOR VALUES OF THE EULER ZETA FUNCTIONS ζE(2n) AT POSITIVE INTEGERS

  • Lee, Hui Young;Ryoo, Cheon Seoung
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.5
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    • pp.1425-1432
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    • 2014
  • The Euler zeta function is defined by ${\zeta}_E(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^8}$. The purpose of this paper is to find formulas of the Euler zeta function's values. In this paper, for $s{\in}\mathbb{N}$ we find the recurrence formula of ${\zeta}_E(2s)$ using the Fourier series. Also we find the recurrence formula of $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{(2_{n-1})^{2s-1}}$, where $s{\geq}2({\in}\mathbb{N})$.

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.

SEVERAL RESULTS ASSOCIATED WITH THE RIEMANN ZETA FUNCTION

  • Choi, Junesang
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.3
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    • pp.467-480
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    • 2009
  • In 1859, Bernhard Riemann, in his epoch-making memoir, extended the Euler zeta function $\zeta$(s) (s > 1; $s{\in}\mathbb{R}$) to the Riemann zeta function $\zeta$(s) ($\Re$(s) > 1; $s{\in}\mathbb{C}$) to investigate the pattern of the primes. Sine the time of Euler and then Riemann, the Riemann zeta function $\zeta$(s) has involved and appeared in a variety of mathematical research subjects as well as the function itself has been being broadly and deeply researched. Among those things, we choose to make a further investigation of the following subjects: Evaluation of $\zeta$(2k) ($k {\in}\mathbb{N}$); Approximate functional equations for $\zeta$(s); Series involving the Riemann zeta function.

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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.

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.