A Quadruple Integral Involving Chebyshev Polynomials Tn(x): Derivation and Evaluation

The aim of the current document is to evaluate a quadruple integral involving the Chebyshev polynomial of the first kind Tn(x) and derive in terms of the Hurwitz-Lerch zeta function. Special cases are evaluated in terms of fundamental constants. The zero distribution of almost all Hurwitz-Lerch zeta functions is asymmetrical. All the results in this work are new.

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A Quadruple Integral Involving Product of the Struve Hv(βt) and Parabolic Cylinder Du(αx) Functions

Symmetry ◽

10.3390/sym14010009 ◽

2021 ◽

Vol 14 (1) ◽

pp. 9

Author(s):

Robert Reynolds ◽

Allan Stauffer

Keyword(s):

Zeta Function ◽

Special Functions ◽

Zeta Functions ◽

Parabolic Cylinder ◽

Fundamental Constants ◽

Lerch Zeta Function ◽

Parabolic Cylinder Functions ◽

Infinite Integral ◽

Almost All

The objective of the present paper is to obtain a quadruple infinite integral. This integral involves the product of the Struve and parabolic cylinder functions and expresses it in terms of the Hurwitz–Lerch Zeta function. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distributionSpecial cases in terms fundamental constants and other special functions are produced. All the results in the work are new.

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A Note on the Summation of the Incomplete Gamma Function

Symmetry ◽

10.3390/sym13122369 ◽

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.

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A Note on a Triple Integral

Symmetry ◽

10.3390/sym13112056 ◽

2021 ◽

Vol 13 (11) ◽

pp. 2056

Author(s):

Robert Reynolds ◽

Allan Stauffer

Keyword(s):

Closed Form ◽

Closed Form Expression ◽

Polynomial Functions ◽

Fundamental Constants ◽

Form Expression ◽

Zero Distribution ◽

Special Cases ◽

Almost All

A closed form expression for a triple integral not previously considered is derived, in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. The kernel of the integral involves the product of the logarithmic, exponential, quotient radical, and polynomial functions. Special cases are derived in terms of fundamental constants; results are summarized in a table. All results in this work are new.

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A Quadruple Definite Integral Expressed in Terms of the Lerch Function

Symmetry ◽

10.3390/sym13091638 ◽

2021 ◽

Vol 13 (9) ◽

pp. 1638

Author(s):

Robert Reynolds ◽

Allan Stauffer

Keyword(s):

Special Functions ◽

Definite Integral ◽

Polynomial Functions ◽

Fundamental Constants ◽

Zero Distribution ◽

Special Cases ◽

Almost All

A quadruple integral involving the logarithmic, exponential and polynomial functions is derived in terms of the Lerch function. Special cases of this integral are evaluated in terms of special functions and fundamental constants. Almost all Lerch functions have an asymmetrical zero-distribution. The majority of the results in this work are new.

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An extension of the generalized Hurwitz-Lerch Zeta function of two variables

Filomat ◽

10.2298/fil1701091c ◽

2017 ◽

Vol 31 (1) ◽

pp. 91-96 ◽

Cited By ~ 4

Author(s):

Junesang Choi ◽

Rakesh Parmar

Keyword(s):

Zeta Function ◽

Zeta Functions ◽

Summation Formula ◽

Integral Representations ◽

Contour Integral ◽

Functions Of Two Variables ◽

Special Cases ◽

Lerch Zeta Function ◽

Function Of Two Variables

The main object of this paper is to introduce a new extension of the generalized Hurwitz-Lerch Zeta functions of two variables. We then systematically investigate such its several interesting properties and related formulas as (for example) various integral representations, which provide certain new and known extensions of earlier corresponding results, a summation formula and Mellin-Barnes type contour integral representations. We also consider some important special cases.

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

10.20944/preprints201803.0008.v1 ◽

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.

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Note on the Hurwitz–Lerch Zeta Function of Two Variables

Symmetry ◽

10.3390/sym12091431 ◽

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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Further Extension of the Generalized Hurwitz-Lerch Zeta Function of Two Variables

Mathematics ◽

10.3390/math7010048 ◽

2019 ◽

Vol 7 (1) ◽

pp. 48

Author(s):

Kottakkaran Sooppy Nisar

Keyword(s):

Zeta Function ◽

Hypergeometric Function ◽

Summation Formula ◽

Integral Representations ◽

Generalized Hypergeometric Function ◽

Special Cases ◽

Lerch Zeta Function ◽

Function Of Two Variables

The main aim of this paper is to provide a new generalization of Hurwitz-Lerch Zeta function of two variables. We also investigate several interesting properties such as integral representations, summation formula, and a connection with the generalized hypergeometric function. To strengthen the main results we also consider some important special cases.

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On a Certain Extension of the Hurwitz-Lerch Zeta Function

Annals of West University of Timisoara - Mathematics and Computer Science ◽

10.2478/awutm-2014-0017 ◽

2014 ◽

Vol 52 (2) ◽

Author(s):

Rakesh K. Parmar ◽

R. K. Raina

Keyword(s):

Zeta Function ◽

Probability Distributions ◽

Integral Representations ◽

Mellin Transforms ◽

Special Cases ◽

Lerch Zeta Function ◽

Generating Relations

AbstractOur purpose in this paper is to consider a generalized form of the extended Hurwitz-Lerch Zeta function. For this extended Hurwitz-Lerch Zeta function, we obtain some classical properties which includes various integral representations, a differential formula, Mellin transforms and certain generating relations. We further consider an application to probability distributions and also point out some important special cases of the main results.

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Real zeros of Hurwitz–Lerch zeta and Hurwitz–Lerch type of Euler–Zagier double zeta functions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004115000547 ◽

2015 ◽

Vol 160 (1) ◽

pp. 39-50 ◽

Cited By ~ 4

Author(s):

TAKASHI NAKAMURA

Keyword(s):

Zeta Function ◽

Zeta Functions ◽

Hurwitz Zeta Function ◽

Real Zeros ◽

Double Zeta ◽

Lerch Zeta Function

AbstractLet 0 < a ⩽ 1, s, z ∈ ${\mathbb{C}}$ and 0 < |z| ⩽ 1. Then the Hurwitz–Lerch zeta function is defined by Φ(s, a, z) ≔ ∑∞n = 0zn(n + a)− s when σ ≔ ℜ(s) > 1. In this paper, we show that the Hurwitz zeta function ζ(σ, a) ≔ Φ(σ, a, 1) does not vanish for all 0 < σ < 1 if and only if a ⩾ 1/2. Moreover, we prove that Φ(σ, a, z) ≠ 0 for all 0 < σ < 1 and 0 < a ⩽ 1 when z ≠ 1. Real zeros of Hurwitz–Lerch type of Euler–Zagier double zeta functions are studied as well.

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