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

Symmetry ◽  
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.

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

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 100
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Chebyshev Polynomial ◽  
Chebyshev Polynomials ◽  
Zeta Functions ◽  
Fundamental Constants ◽  
Zero Distribution ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Almost All

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.

scholarly journals Definite Integrals Involving Combinations of Powers and Logarithmic Functions of Complicated Arguments Expressed in Terms of the Hurwitz Zeta Function

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Zeta Function ◽  
Special Functions ◽  
Zeta Functions ◽  
Hurwitz Zeta Function ◽  
Fundamental Constants ◽  
Closed Formula ◽  
Definite Integrals ◽  
Logarithmic Functions

In this manuscript, the authors derive closed formula for definite integrals of combinations of powers and logarithmic functions of complicated arguments and express these integrals in terms of the Hurwitz zeta functions. These derivations are then expressed in terms of fundamental constants, elementary, and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.

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 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.

An infinite integral involving Bessel functions and parabolic cylinder functions

1937 ◽  
Vol 33 (2) ◽  
pp. 210-211 ◽  
Author(s):  
R. S. Varma
Keyword(s):  
Bessel Functions ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Infinite Integral

The object of this paper is to evaluate an infinite integral involving Bessel functions and parabolic cylinder functions. The following two lemmas are required:Lemma 1. provided that R(m) > 0.

scholarly journals Distributions of the Ratio and Product of Two Independent Weibull and Lindley Random Variables

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
N. J. Hassan ◽  
A. Hawad Nasar ◽  
J. Mahdi Hadad
Keyword(s):  
Hypergeometric Functions ◽  
Special Functions ◽  
Random Variables ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Parabolic Cylinder ◽  
Generalized Hypergeometric Functions ◽  
Confluent Hypergeometric Functions ◽  
Parabolic Cylinder Functions ◽  
The Moment

In this paper, we derive the cumulative distribution functions (CDF) and probability density functions (PDF) of the ratio and product of two independent Weibull and Lindley random variables. The moment generating functions (MGF) and the k-moment are driven from the ratio and product cases. In these derivations, we use some special functions, for instance, generalized hypergeometric functions, confluent hypergeometric functions, and the parabolic cylinder functions. Finally, we draw the PDF and CDF in many values of the parameters.

scholarly journals A Quadruple Definite Integral Expressed in Terms of the Lerch Function

Symmetry ◽  
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.

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

2015 ◽  
Vol 160 (1) ◽  
pp. 39-50 ◽  
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.

scholarly journals Local zeta functions and Newton polyhedra

2003 ◽  
Vol 172 ◽  
pp. 31-58 ◽  
Author(s):  
W. A. Zuniga-Galindo
Keyword(s):  
Zeta Function ◽  
Valuation Ring ◽  
Zeta Functions ◽  
Newton Polyhedron ◽  
Group Of Units ◽  
Local Zeta Function ◽  
Degenerate Polynomial ◽  
Small Set ◽  
Local Zeta Functions ◽  
Almost All

AbstractTo a polynomial f over a non-archimedean local field K and a character χ of the group of units of the valuation ring of K one associates Igusa’s local zeta function Z (s, f, χ). In this paper, we study the local zeta function Z(s, f, χ) associated to a non-degenerate polynomial f, by using an approach based on the p-adic stationary phase formula and Néron p-desingularization. We give a small set of candidates for the poles of Z (s, f, χ) in terms of the Newton polyhedron Γ(f) of f. We also show that for almost all χ, the local zeta function Z(s, f, χ) is a polynomial in q−s whose degree is bounded by a constant independent of χ. Our second result is a description of the largest pole of Z(s, f, χtriv) in terms of Γ(f) when the distance between Γ(f) and the origin is at most one.

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