Related papers: Explicit formula for generalization of Poly-Bernou…
A generalization of a well-known relation between the Riemann zeta function $\zeta(s)$ and Bernoulli numbers $B_n$ is obtained. The formula is a new representation of the Riemann zeta function in terms of a nested series of Bernoulli…
We characterize the generalized Chebyshev polynomials of the second kind (Chebyshev-II), and then we provide a closed form of the generalized Chebyshev-II polynomials using the Bernstein basis. These polynomials can be used to describe the…
For a two parameter family of Bernoulli numbers $B_{n, p}$ the exponential generating function is derived by elementary methods.
We deduce Katz's theorems for $(A,B)$-exponential sums over finite fields using $\ell$-adic cohomology and a theorem of Denef-Loeser, removing the hypothesis that $A+B$ is relatively prime to the characteristic $p$. In some degenerate…
In this paper, we give various identities for the weighted average of the product of generalized Anderson-Apostol sums with weights concerning completely multiplicative function, completely additive function, logarithms, the Gamma function,…
It is known that the special values of multiple zeta functions at non-positive arguments are indeterminate in most cases due to the occurrences of infinitely many singularities. In order to give a suitable rigorous meaning of the special…
We present a new explicit formula for the $m$-th Bernoulli number $B_m$, which involves two integer parameters $a$ and $n$ with $0\le a\le m\le n$. If we set $a=0$ and $n=m$, then the formula reduces to the celebrated Kronecker formula for…
Here we consider the degenerate Bernstein polynomials as a degenerate version of Bernstein polynomials, which are motivated by Simsek's recent work 'Generating functions for unification of the multidimensional Bernstein polynomials and…
In this paper, we consider the poly-Bernoulli numbers and polynomials of the second kind and presents new and explicit formulae for calculating the poly-Bernoulli numbers of the second kind and the Stirling numbers of the second kind.
In this paper, a new form of poly-Genocchi polynomials is defined by means of poly-logarithm, namely, the Apostol-type poly-Genocchi polynomials of higher order with parameters a, b and c. Several properties of these polynomials are…
In a previous paper, Rahmani introduced a new family of p-Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. Motivated by this paper and as a degenerate version of those numbers and polynomials, we introduce…
This is the second part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. In this part, we give constructive results generalizing previous ones obtained by the author in…
We introduce generalized hypergeometric Bernoulli numbers for Dirichlet characters. We study their properties, including relations, expressions and determinants. At the end in Appendix we derive first few expressions of these numbers.
In this paper, we introduce a novel identity for generalized Euler polynomials, leading to further generalizations for several relations involving classical Euler numbers, Euler polynomials, Genocchi polynomials, and Genocchi numbers.
We provide a multidimensional weighted Euler--MacLaurin summation formula on polytopes and a multidimensional generalization of a result due to L. J. Mordell on the series expansion in Bernoulli polynomials. These results are consequences…
We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on $[-1,1]$. The recurrence coefficients…
In this paper, the problem of multiplicative anomaly of zeta regularization is solved for polynomials. For a regularizable sequence $\Lambda$, we explicitly calculate the zeta regularized product of $(\Lambda-z_1)\dots(\Lambda-z_n)$ for…
Kaneko and Tsumura introduced the Arakawa-Kaneko type zeta function $\eta(-k_1,\ldots,-k_r;s_1,\ldots,s_r)$ for non-negative integers $k_1,\ldots,k_r$ and complex variables $s_1,\ldots,s_r$. Recently, Yamamoto showed that, by using the…
The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials {n} in variables s and t given by {0} = 0, {1} = 1, and {n} = s{n-1}+t{n-2} for n ge 2. The latter are defined…
We explore a variant of the zeta function interpolating the Bernoulli numbers based on an integral representation suggested by J. Jensen. The Bernoulli function $\operatorname{B}(s, v) = - s\, \zeta(1-s, v)$ can be introduced independently…