Bernoulli numbers and zeta functions

2015 ◽  
Vol 52 (10) ◽  
pp. 52-5384-52-5384
Keyword(s):  
Zeta Functions ◽  
Bernoulli Numbers

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 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.

THE ERROR ZETA FUNCTION

2007 ◽  
Vol 03 (03) ◽  
pp. 439-453 ◽  
Author(s):  
ABDUL HASSEN ◽  
HIEU D. NGUYEN
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Special Function ◽  
Zeta Functions ◽  
Bernoulli Numbers ◽  
Intimate Connection

This paper investigates a new special function referred to as the error zeta function. Derived as a fractional generalization of hypergeometric zeta functions, the error zeta function is shown to exhibit many properties analogous to its hypergeometric counterpart, including its intimate connection to Bernoulli numbers. These new properties are treated in detail and used to demonstrate a pre-functional equation satisfied by this special function.

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.

ON A WEIGHTED SUM OF MULTIPLE -VALUES OF FIXED WEIGHT AND DEPTH

2021 ◽  
pp. 1-8
Author(s):  
YOSHIHIRO TAKEYAMA
Keyword(s):  
Differential Equation ◽  
Zeta Functions ◽  
Bernoulli Numbers ◽  
Weighted Sum ◽  
Multiple Zeta Values ◽  
Multiple Zeta Value ◽  
Fixed Weight ◽  
Multiple Zeta ◽  
Pure Mathematics ◽  
Zeta Values

AbstractThe multipleT-value, which is a variant of the multiple zeta value of level two, was introduced by Kaneko and Tsumura [‘Zeta functions connecting multiple zeta values and poly-Bernoulli numbers’, in:Various Aspects of Multiple Zeta Functions, Advanced Studies in Pure Mathematics, 84 (Mathematical Society of Japan, Tokyo, 2020), 181–204]. We show that the generating function of a weighted sum of multipleT-values of fixed weight and depth is given in terms of the multipleT-values of depth one by solving a differential equation of Heun type.

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