Related papers: Some results associated with Bernoulli and Euler n…
We provide several simple recursive formulae for the moment sequence of infinite Bernoulli convolution. We relate moments of one infinite Bernoulli convolution with others having different but related parameters. We give examples relating…
We provide an explicit formula for the Tornheim double series T(a,0,c) in terms of an integral involving the Hurwitz zeta function. For integer values of the parameters, a=m, c=n, we show that in the most interesting case of even weight…
In the present paper we introduce some expansions, based on the falling factorials, for the Euler Gamma function and the Riemann Zeta function. In the proofs we use the Fa\'a di Bruno formula, Bell polynomials, potential polynomials,…
We define a parametric variant of generalized Euler sums and construct contour integration to give some explicit evaluations of these parametric Euler sums. In particular, we establish several explicit formulas of (Hurwitz) zeta functions,…
The aim of this paper is to find the numerical solutions of the second order linear and nonlinear differential equations with Dirichlet, Neumann and Robin boundary conditions. We use the Bernoulli polynomials as linear combination to the…
In this paper, we investigate new class of sequences related to fully degenerate Bernoulli numbers and polynomials. From those sequences, we derive some formulae for the degenerate Bernoulli and Euler polynomials.
We present another expression to regularize the Euler product representation of the Riemann zeta function. % in this paper. The expression itself is essentially same as the usual Euler product that is the infinite product, but we define a…
In this paper new series for the first and second Stieltjes constants (also known as generalized Euler's constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the…
In this paper, the power series and hypergeometric series representations of the beta and Ramanujan functions \begin{equation*} \mathcal{B}\left( x\right) =\frac{\Gamma \left( x\right)^{2}}{\Gamma \left( 2x\right) }\text{ and…
Let $$ \zeta_E(s,q)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+q)^{s}} $$ be the alternating Hurwitz (or Hurwitz-type Euler) zeta function. In this paper, we obtain the following asymptotic expansion of $\zeta_{E}(s,q)$ $$ \zeta_E(s,q)\sim\frac12…
This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers $H_{n}^{\left( p,q\right) }$ \[ \zeta_{H^{\left( p,q\right) }}\left( r\right) =\sum\limits_{n=1}^{\infty }\dfrac{H_{n}^{\left( p,q\right) }}{n^{r}}%…
In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) $\gamma_m$ are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in $\pi^{-2}$ with rational…
In this paper, we establish an extension of a noncommutative Bennett inequality with a parameter $1\leq r\leq2$ and use it together with some noncommutative techniques to establish a Rosenthal inequality. We also present a noncommutative…
Linear harmonic number sums had been studied by a variety of authors during the last centuries, but only few results are known about nonlinear Euler sums of quadratic or even higher degree. The first systematic study on nonlinear Euler sums…
The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application…
For a set $A$ of non-negative integers, let $R_A(n)$ denote the number of solutions to the equation $n=a+a'$ with $a$, $a'\in A$. Denote by $\chi_A(n)$ the characteristic function of $A$. Let $b_n>0$ be a sequence satisfying $\limsup_{n\to…
We consider the numbers $\mathcal{B}_{r,s} = (\mathbf{B}+1)^r \mathbf{B}^s$ (in umbral notation $\mathbf{B}^n = \mathbf{B}_n$ with the Bernoulli numbers) that have a well-known reciprocity relation, which is frequently found in the…
It is well known that $\zeta(2)=\pi^2/6$ as discovered by Euler. In this paper we present the following two $q$-analogues of this celebrated formula:…
Let $\alpha = (1+\sqrt{5})/2$ and define the lower and upper Wythoff sequences by $a_i = \lfloor i \alpha \rfloor$, $b_i = \lfloor i \alpha^2 \rfloor$ for $i \geq 1$. In a recent interesting paper, Kawsumarng et al. proved a number of…
This is the second paper in a series where we study arithmetic applications of the multiple elliptic Gamma functions originated in mathematical physics. In the first article in this series we defined geometric families of these functions…