scholarly journals Multiple Dedekind Type Sums and Their Related Zeta Functions

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1744
Author(s):  
Abdelmejid Bayad ◽  
Yilmaz Simsek
Keyword(s):  
Zeta Functions ◽  
Bernoulli Polynomials ◽  
Reciprocity Laws ◽  
Bernoulli Functions ◽  
Interpolation Functions

The main purpose of this paper is to use the multiple twisted Bernoulli polynomials and their interpolation functions to construct multiple twisted Dedekind type sums. We investigate some properties of these sums. By use of the properties of multiple twisted zeta functions and the Bernoulli functions involving the Bernoulli polynomials, we derive reciprocity laws of these sums. Further developments and observations on these new Dedekind type sums are given.

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.

ON INTERPOLATION FUNCTIONS FOR GENERALIZED Li COEFFICIENTS IN THE SELBERG CLASS

2011 ◽  
Vol 07 (03) ◽  
pp. 771-792
Author(s):  
ALMASA ODŽAK ◽  
LEJLA SMAJLOVIĆ
Keyword(s):  
Exponential Type ◽  
Entire Functions ◽  
Complex Function ◽  
Zeta Functions ◽  
Unitary Representations ◽  
Selberg Class ◽  
Interpolation Function ◽  
Entire Complex ◽  
Class Of Functions ◽  
Interpolation Functions

We prove that there exists an entire complex function of order one and finite exponential type that interpolates the Li coefficients λF(n) attached to a function F in the class [Formula: see text] that contains both the Selberg class of functions and (unconditionally) the class of all automorphic L-functions attached to irreducible, cuspidal, unitary representations of GL n(ℚ). We also prove that the interpolation function is (essentially) unique, under generalized Riemann hypothesis. Furthermore, we obtain entire functions of order one and finite exponential type that interpolate both archimedean and non-archimedean contribution to λF(n) and show that those functions can be interpreted as zeta functions built, respectively, over trivial zeros and all zeros of a function [Formula: see text].

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.

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