AN INTEGRAL REPRESENTATION OF MULTIPLE HURWITZ-LERCH ZETA FUNCTIONS AND GENERALIZED MULTIPLE BERNOULLI NUMBERS

2009 ◽  
Vol 61 (4) ◽  
pp. 437-496 ◽  
Author(s):  
Y. Komori
Keyword(s):  
Integral Representation ◽  
Zeta Functions ◽  
Bernoulli Numbers

scholarly journals HYPERGEOMETRIC ZETA FUNCTIONS

2010 ◽  
Vol 06 (01) ◽  
pp. 99-126 ◽  
Author(s):  
ABDUL HASSEN ◽  
HIEU D. NGUYEN
Keyword(s):  
Integral Representation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Special Functions ◽  
Zeta Functions ◽  
Functional Inequality ◽  
Bernoulli Numbers ◽  
Classical Counterpart ◽  
New Family ◽  
Riemann Zeta

This paper investigates a new family of special functions referred to as hypergeometric zeta functions. Derived from the integral representation of the classical Riemann zeta function, hypergeometric zeta functions exhibit many properties analogous to their classical counterpart, including the intimate connection to Bernoulli numbers. These new properties are treated in detail and are used to demonstrate a functional inequality satisfied by second-order hypergeometric zeta functions.

scholarly journals Fractional calculus, zeta functions and Shannon entropy

2021 ◽  
Vol 19 (1) ◽  
pp. 87-100
Author(s):  
Emanuel Guariglia
Keyword(s):  
Functional Equation ◽  
Fractional Calculus ◽  
Integral Representation ◽  
Fractional Derivative ◽  
Shannon Entropy ◽  
Zeta Functions ◽  
Bernoulli Numbers

Abstract This paper deals with the fractional calculus of zeta functions. In particular, the study is focused on the Hurwitz ζ \zeta function. All the results are based on the complex generalization of the Grünwald-Letnikov fractional derivative. We state and prove the functional equation together with an integral representation by Bernoulli numbers. Moreover, we treat an application in terms of Shannon entropy.

p-adic zeta functions and Bernoulli numbers

1982 ◽  
Vol 19 (2) ◽  
pp. 1186-1194 ◽  
Author(s):  
Yu. V. Osipov
Keyword(s):  
Zeta Functions ◽  
Bernoulli Numbers

Bernoulli Numbers and Zeta Functions

2014 ◽  
Author(s):  
Tsuneo Arakawa ◽  
Tomoyoshi Ibukiyama ◽  
Masanobu Kaneko
Keyword(s):  
Zeta Functions ◽  
Bernoulli Numbers

scholarly journals Some Difference Equations for Srivastava’s λ-Generalized Hurwitz–Lerch Zeta Functions with Applications

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 311 ◽  
Author(s):  
Asifa Tassaddiq
Keyword(s):  
Integral Representation ◽  
Difference Equations ◽  
Zeta Functions ◽  
Numerical Computations ◽  
Direct Evaluation ◽  
Mellin Transforms ◽  
The Family

In this article, we establish some new difference equations for the family of λ-generalized Hurwitz–Lerch zeta functions. These difference equations proved worthwhile to study these newly defined functions in terms of simpler functions. Several authors investigated such functions and their analytic properties, but no work has been reported for an estimation of their values. We perform some numerical computations to evaluate these functions for different values of the involved parameters. It is shown that the direct evaluation of involved integrals is not possible for the large values of parameter s; nevertheless, using our new difference equations, we can evaluate these functions for the large values of s. It is worth mentioning that for the small values of this parameter, our results are 100% accurate with the directly computed results using their integral representation. Difference equations so obtained are also useful for the computation of some new integrals of products of λ-generalized Hurwitz–Lerch zeta functions and verified to be consistent with the existing results. A derivative property of Mellin transforms proved fundamental to present this investigation.

scholarly journals Some Identities on theq-Bernoulli Numbers and Polynomials with Weight 0

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
T. Kim ◽  
J. Choi ◽  
Y. H. Kim
Keyword(s):  
Integral Representation ◽  
Bernstein Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials

Recently, Kim (2011) has introduced theq-Bernoulli numbers with weightα. In this paper, we consider theq-Bernoulli numbers and polynomials with weightα=0and givep-adicq-integral representation of Bernstein polynomials associated withq-Bernoulli numbers and polynomials with weight0. From these integral representation onℤp, we derive some interesting identities on theq-Bernoulli numbers and polynomials with weight0.

scholarly journals On p-adic Integral Representation of q-Bernoulli Numbers Arising from Two Variable q-Bernstein Polynomials

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 451 ◽  
Author(s):  
Dae Kim ◽  
Taekyun Kim ◽  
Cheon Ryoo ◽  
Yonghong Yao
Keyword(s):  
Integral Representation ◽  
Bernstein Polynomials ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Finite Product ◽  
Bernoulli Numbers And Polynomials ◽  
Special Values ◽  
Invariant Integrals ◽  
Evaluation Problem ◽  
Double Integrals

The q-Bernoulli numbers and polynomials can be given by Witt’s type formulas as p-adic invariant integrals on Z p . We investigate some properties for them. In addition, we consider two variable q-Bernstein polynomials and operators and derive several properties for these polynomials and operators. Next, we study the evaluation problem for the double integrals on Z p of two variable q-Bernstein polynomials and show that they can be expressed in terms of the q-Bernoulli numbers and some special values of q-Bernoulli polynomials. This is generalized to the problem of evaluating any finite product of two variable q-Bernstein polynomials. Furthermore, some identities for q-Bernoulli numbers are found.

scholarly journals On the Values of the Weighted -Zeta and -Functions

2011 ◽  
Vol 2011 ◽  
pp. 1-7
Author(s):  
T. Kim ◽  
S. H. Lee ◽  
Hyeon-Ho Han ◽  
C. S. Ryoo
Keyword(s):  
Zeta Functions ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Weighted Functions

Recently, the modified -Bernoulli numbers and polynomials are introduced in (D. V. Dolgy et al., in press). These numbers are valuable to study the weighted -zeta and -functions. In this paper, we study the weighted -zeta functions and weighted -functions from the modified -Bernoulli numbers and polynomials with weight .

scholarly journals Analytical Solution to Improper Integral of Divergent Power Functions Using The Riemann Zeta Function

2019 ◽  
Author(s):  
Siamak Tafazoli ◽  
Farhad Aghili
Keyword(s):  
Closed Form Solution ◽  
Zeta Functions ◽  
Bernoulli Numbers ◽  
Form Solution ◽  
Binomial Coefficients ◽  
Solution Technique ◽  
Power Functions ◽  
Definite Integrals ◽  
Improper Integral ◽  
Riemann Zeta

This paper presents an analytical closed-form solution to improper integral $\mu(r)=\int_0^{\infty} x^r dx$, where $r \geq 0$. The solution technique is based on splitting the improper integral into an infinite sum of definite integrals with successive integer limits. The exact solution of every definite integral is obtained by making use of the binomial polynomial expansion, which then allows expression of the entire summation equivalently in terms of a weighted sum of Riemann zeta functions. It turns out that the solution fundamentally depends on whether or not $r$ is an integer. If $r$ is a non-negative integer, then the solution is manifested in a finite series of weighted Bernoulli numbers, which is then drastically simplified to a second order rational function $\mu(r)=(-1)^{r+1}/(r+1)(r+2)$. This is achieved by taking advantage of the relationships between Bernoulli numbers and binomial coefficients. On the other hand, if $r$ is a non-integer real-valued number, then we prove $\mu(r)=0$ by the virtue of the elegant relationships between zeta and gamma functions and their properties.

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