The theory of Bernoulli polynomials via zeta-functions

2009 ◽  
pp. 77-80
Keyword(s):  
Zeta Functions ◽  
Bernoulli Polynomials

scholarly journals Two types of hypergeometric degenerate Cauchy numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 417-433
Author(s):  
Takao Komatsu
Keyword(s):  
Zeta Functions ◽  
Bernoulli Polynomials ◽  
Cauchy Numbers

Abstract In 1985, Howard introduced degenerate Cauchy polynomials together with degenerate Bernoulli polynomials. His degenerate Bernoulli polynomials have been studied by many authors, but his degenerate Cauchy polynomials have been forgotten. In this paper, we introduce some kinds of hypergeometric degenerate Cauchy numbers and polynomials from the different viewpoints. By studying the properties of the first one, we give their expressions and determine the coefficients. Concerning the second one, called H-degenerate Cauchy polynomials, we show several identities and study zeta functions interpolating these polynomials.

Bernoulli and poly-Bernoulli polynomial convolutions and identities of p-adic Arakawa–Kaneko zeta functions

2016 ◽  
Vol 12 (05) ◽  
pp. 1295-1309 ◽  
Author(s):  
Paul Thomas Young
Keyword(s):  
Closed Form ◽  
Zeta Functions ◽  
Bernoulli Polynomials ◽  
Bernoulli Polynomial ◽  
Sum Formula

We evaluate the ordinary convolution of Bernoulli polynomials in closed form in terms of poly-Bernoulli polynomials. As applications we derive identities for [Formula: see text]-adic Arakawa–Kaneko zeta functions, including a [Formula: see text]-adic analogue of Ohno’s sum formula. These [Formula: see text]-adic identities serve to illustrate the relationships between real periods and their [Formula: see text]-adic analogues.

scholarly journals Identities for Bernoulli polynomials related to multiple Tornheim zeta functions

2019 ◽  
Vol 476 (2) ◽  
pp. 569-584 ◽  
Author(s):  
Karl Dilcher ◽  
Armin Straub ◽  
Christophe Vignat
Keyword(s):  
Zeta Functions ◽  
Bernoulli Polynomials

scholarly journals Relations for Bernoulli–Barnes numbers and Barnes zeta functions

2014 ◽  
Vol 10 (05) ◽  
pp. 1321-1335 ◽  
Author(s):  
Abdelmejid Bayad ◽  
Matthias Beck
Keyword(s):  
Generating Function ◽  
Zeta Functions ◽  
Bernoulli Polynomials

The Barnes ζ-function is [Formula: see text] defined for [Formula: see text], Re (x) > 0, and Re (z) > n and continued meromorphically to ℂ. Specialized at negative integers -k, the Barnes ζ-function gives [Formula: see text] where Bk(x; a) is a Bernoulli–Barnes polynomial, which can be also defined through a generating function that has a slightly more general form than that for Bernoulli polynomials. Specializing Bk(0; a) gives the Bernoulli–Barnes numbers. We exhibit relations among Barnes ζ-functions, Bernoulli–Barnes numbers and polynomials, which generalize various identities of Agoh, Apostol, Dilcher, and Euler.

scholarly journals A Newq-Analogue of Bernoulli Polynomials Associated withp-Adicq-Integrals

2008 ◽  
Vol 2008 ◽  
pp. 1-6 ◽  
Author(s):  
Lee-Chae Jang
Keyword(s):  
Zeta Functions ◽  
Bernoulli Polynomials

We will study a newq-analogue of Bernoulli polynomials associated withp-adicq-integrals. Furthermore, we examine the Hurwitz-typeq-zeta functions, replacingp-adic rational integersxwith aq-analogue[x]qfor ap-adic numberqwith|q−1|p<1, which interpolateq-analogue of Bernoulli polynomials.

scholarly journals On Generalized Arakawa–Kaneko Zeta Functions with Parameters a,b,c

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Nestor G. Acala ◽  
Edward Rowe M. Aleluya
Keyword(s):  
Zeta Functions ◽  
Bernoulli Polynomials ◽  
Interpolation Formula

For k∈ℤ, the generalized Arakawa–Kaneko zeta functions with a, b, c parameters are given by the Laplace-Mellin integral ξks,x;a,b,c=1/Γs∫0∞Lik1−ab−t/bt−a−tc−xtts−1dt, where ℜs>0 and x>0 if k≥1, and ℜs>0 and x>k+1 if k≤0. In this paper, an interpolation formula between these generalized zeta functions and the poly-Bernoulli polynomials with a,b,c parameters is obtained. Moreover, explicit, difference, and Raabe’s formulas for ξks,x;a,b,c are derived.

Recursion formulas for poly-Bernoulli numbers and their applications

2020 ◽  
pp. 1-15
Author(s):  
Yasuo Ohno ◽  
Yoshitaka Sasaki
Keyword(s):  
Zeta Functions ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Positive Answer ◽  
Recurrence Formulas ◽  
Recursion Formulas ◽  
Counting Formula

Recurrence formulas for generalized poly-Bernoulli polynomials are given. The formula gives a positive answer to a question raised by Kaneko. Further, as applications, annihilation formulas for Arakawa-Kaneko type zeta-functions and a counting formula for lonesum matrices of a certain type are also discussed.

scholarly journals Multivariatep-Adic Fermionicq-Integral onℤpand Related Multiple Zeta-Type Functions

2008 ◽  
Vol 2008 ◽  
pp. 1-13 ◽  
Author(s):  
Min-Soo Kim ◽  
Taekyun Kim ◽  
Jin-Woo Son
Keyword(s):  
Generating Functions ◽  
Zeta Functions ◽  
Bernoulli Polynomials ◽  
Partial Answer ◽  
Euler Polynomials ◽  
Multiple Zeta

In 2008, Jang et al. constructed generating functions of the multiple twisted Carlitz's typeq-Bernoulli polynomials and obtained the distribution relation for them. They also raised the following problem:“are there analytic multiple twisted Carlitz's typeq-zeta functions which interpolate multiple twisted Carlitz's typeq-Euler (Bernoulli) polynomials?”The aim of this paper is to give a partial answer to this problem. Furthermore we derive some interesting identities related to twistedq-extension of Euler polynomials and multiple twisted Carlitz's typeq-Euler polynomials.

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