• Title/Summary/Keyword: Hurwitz zeta function

Search Result 49, Processing Time 0.022 seconds

CERTAIN NEW EXTENSION OF HURWITZ-LERCH ZETA FUNCTION

  • KHAN, WASEEM A.;GHAYASUDDIN, M.;AHMAD, MOIN
    • Journal of applied mathematics & informatics
    • /
    • v.37 no.1_2
    • /
    • pp.13-21
    • /
    • 2019
  • In the present research paper, we introduce a further extension of Hurwitz-Lerch zeta function by using the generalized extended Beta function defined by Parmar et al.. We investigate its integral representations, Mellin transform, generating functions and differential formula. In view of diverse applications of the Hurwitz-Lerch Zeta functions, the results presented here may be potentially useful in some related research areas.

INCOMPLETE EXTENDED HURWITZ-LERCH ZETA FUNCTIONS AND ASSOCIATED PROPERTIES

  • Parmar, Rakesh K.;Saxena, Ram K.
    • Communications of the Korean Mathematical Society
    • /
    • v.32 no.2
    • /
    • pp.287-304
    • /
    • 2017
  • Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [Integral Transforms Spec. Funct. 23 (2012), 659-683] by means of the incomplete Pochhammer symbols $({\lambda};{\kappa})_{\nu}$ and $[{\lambda};{\kappa}]_{\nu}$, we first introduce incomplete Fox-Wright function. We then define the families of incomplete extended Hurwitz-Lerch Zeta function. We then systematically investigate several interesting properties of these incomplete extended Hurwitz-Lerch Zeta function which include various integral representations, summation formula, fractional derivative formula. We also consider an application to probability distributions and some special cases of our main results.

On Extended Hurwitz-Lerch Zeta Function

  • Mohannad Jamal Said Shahwan;Maged Gumman Bin-Saad;Mohammed Ahmed Pathan
    • Kyungpook Mathematical Journal
    • /
    • v.63 no.3
    • /
    • pp.485-506
    • /
    • 2023
  • This paper investigates an extended form Hurwitz-Lerch zeta function, as well as related integral images, ordinary and fractional derivatives, and series expansions, using the term extended beta function. We establish a connection between the extended Hurwitz-Lerch zeta function and the Laguerre polynomials. Furthermore, we present a probability distribution application of the extended Hurwitz-Lerch zeta function ζ𝛿,𝜇𝜈,λ. Several results, both known and new, are shown to follow as special cases of our findings.

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
    • /
    • v.55 no.1
    • /
    • pp.185-210
    • /
    • 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 EXTENSION OF THE EXTENDED HURWITZ-LERCH ZETA FUNCTIONS OF TWO VARIABLES

  • Choi, Junesang;Parmar, Rakesh K.;Saxena, Ram K.
    • Bulletin of the Korean Mathematical Society
    • /
    • v.54 no.6
    • /
    • pp.1951-1967
    • /
    • 2017
  • We aim to introduce a further extension of a family of the extended Hurwitz-Lerch Zeta functions of two variables. We then systematically investigate several interesting properties of the extended function such as its integral representations which provide extensions of various earlier corresponding results of two and one variables, its summation formula, its Mellin-Barnes type contour integral representations, its computational representation and fractional derivative formulas. A multi-parameter extension of the extended Hurwitz-Lerch Zeta function of two variables is also introduced. Relevant connections of certain special cases of the main results presented here with some known identities are pointed out.

THE COMPOSITION OF HURWITZ-LERCH ZETA FUNCTION WITH PATHWAY INTEGRAL OPERATOR

  • Jangid, Nirmal Kumar;Joshi, Sunil;Purohit, Sunil Dutt;Suthar, Daya Lal
    • Communications of the Korean Mathematical Society
    • /
    • v.36 no.2
    • /
    • pp.267-276
    • /
    • 2021
  • The aim of the present investigation is to establish the composition formulas for the pathway fractional integral operator connected with Hurwitz-Lerch zeta function and extended Wright-Bessel function. Some interesting special cases have also been discussed.

Extension of Generalized Hurwitz-Lerch Zeta Function and Associated Properties

  • Choi, Junesang;Parmar, Rakesh Kumar;Raina, Ravinder Krishna
    • Kyungpook Mathematical Journal
    • /
    • v.57 no.3
    • /
    • pp.393-400
    • /
    • 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.

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

  • MIN-SOO KIM
    • Journal of applied mathematics & informatics
    • /
    • v.41 no.6
    • /
    • pp.1409-1418
    • /
    • 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).