scholarly journals The Lerch Zeta function III. Polylogarithms and special values

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Jeffrey C. Lagarias ◽  
Wen-Ching Winnie Li
Keyword(s):  
Zeta Function ◽  
Special Values ◽  
Lerch Zeta Function

scholarly journals Series Representations at Special Values of Generalized Hurwitz-Lerch Zeta Function

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
S. Gaboury ◽  
A. Bayad
Keyword(s):  
Zeta Function ◽  
Higher Order ◽  
Euler Polynomials ◽  
Expansion Formula ◽  
Special Values ◽  
Lerch Zeta Function ◽  
Series Representations ◽  
Rational Arguments

By making use of some explicit relationships between the Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi, and Apostol-Frobenius-Euler polynomials of higher order and the generalized Hurwitz-Lerch zeta function as well as a new expansion formula for the generalized Hurwitz-Lerch zeta function obtained recently by Gaboury and Bayad , in this paper we present some series representations for these polynomials at rational arguments. These results provide extensions of those obtained by Apostol (1951) and by Srivastava (2000).

SPECIAL VALUES OF SOME EXTENSIONS OF ZETA FUNCTION VIA BETA DISTRIBUTIONS

2012 ◽  
Vol 15 (02) ◽  
pp. 1250008
Author(s):  
NORIYOSHI SAKUMA ◽  
RYOICHI SUZUKI
Keyword(s):  
Zeta Function ◽  
Poisson Distribution ◽  
Special Values ◽  
Beta Distributions ◽  
Lerch Zeta Function ◽  
Basel Problem

In this paper, we show the Basel problem via the beta distributions, which include the free Poisson distribution and positive arcsine law. This is a generalization of Ref. 2 by Fujita. We also obtain special values of an extension of generalized Hurwitz–Lerch zeta function, which was introduced by Gang, Jain and Kalla.

scholarly journals A Note on the Summation of the Incomplete Gamma Function

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2369
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Gamma Function ◽  
Zeta Functions ◽  
Incomplete Gamma Function ◽  
Zero Distribution ◽  
Special Values ◽  
Lerch Zeta Function ◽  
Almost All ◽  
Infinite Sum

We examine the improved infinite sum of the incomplete gamma function for large values of the parameters involved. We also evaluate the infinite sum and equivalent Hurwitz-Lerch zeta function at special values and produce a table of results for easy reading. Almost all Hurwitz-Lerch zeta functions have an asymmetrical zero distribution.

scholarly journals Certain Integral Operator Related to the Hurwitz–Lerch Zeta Function

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Xiao-Yuan Wang ◽  
Lei Shi ◽  
Zhi-Ren Wang
Keyword(s):  
Integral Operator ◽  
Zeta Function ◽  
Meromorphic Functions ◽  
Third Order ◽  
Lerch Zeta Function ◽  
Sandwich Type ◽  
Differential Superordination ◽  
Differential Subordinations

The aim of the present paper is to investigate several third-order differential subordinations, differential superordination properties, and sandwich-type theorems of an integral operator Ws,bf(z) involving the Hurwitz–Lerch Zeta function. We make some applications of the operator Ws,bf(z) for meromorphic functions.

scholarly journals Two-sided inequalities for the extended Hurwitz–Lerch Zeta function

2011 ◽  
Vol 62 (1) ◽  
pp. 516-522 ◽  
Author(s):  
H.M. Srivastava ◽  
Dragana Jankov ◽  
Tibor K. Pogány ◽  
R.K. Saxena
Keyword(s):  
Zeta Function ◽  
Lerch Zeta Function

scholarly journals A (p,v)-Extension of Hurwitz-Lerch Zeta Function and its Properties

2018 ◽  
Author(s):  
Gauhar Rahman ◽  
KS Nisar ◽  
Shahid Mubeen
Keyword(s):  
Zeta Function ◽  
Beta Function ◽  
Integral Representations ◽  
Basic Properties ◽  
Mellin Transformation ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Derivative Formulas ◽  
Generating Relations

In this paper, we define a (p,v)-extension of Hurwitz-Lerch Zeta function by considering an extension of beta function defined by Parmar et al. [J. Classical Anal. 11 (2017) 81–106]. We obtain its basic properties which include integral representations, Mellin transformation, derivative formulas and certain generating relations. Also, we establish the special cases of the main results.

scholarly journals Note on the Hurwitz–Lerch Zeta Function of Two Variables

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1431
Author(s):  
Junesang Choi ◽  
Recep Şahin ◽  
Oğuz Yağcı ◽  
Dojin Kim
Keyword(s):  
Zeta Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Zeta Functions ◽  
Integral Representations ◽  
Lerch Zeta Function ◽  
Derivative Formulas ◽  
The One ◽  
Function Of Two Variables

A number of generalized Hurwitz–Lerch zeta functions have been presented and investigated. In this study, by choosing a known extended Hurwitz–Lerch zeta function of two variables, which has been very recently presented, in a systematic way, we propose to establish certain formulas and representations for this extended Hurwitz–Lerch zeta function such as integral representations, generating functions, derivative formulas and recurrence relations. We also point out that the results presented here can be reduced to yield corresponding results for several less generalized Hurwitz–Lerch zeta functions than the extended Hurwitz–Lerch zeta function considered here. For further investigation, among possibly various more generalized Hurwitz–Lerch zeta functions than the one considered here, two more generalized settings are provided.

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