Related papers: Asymptotics and exact formulas for Zagier polynomi…
By using Cauchy's formula, it is known that Bernoulli numbers and Euler numbers can be represented by the contour integrals \begin{equation*} \begin{aligned} B_n&=\frac{n!}{2\pi i}\oint \frac{z}{e^z-1}\frac{d…
We present various identities involving the classical Bernoulli and Euler polynomials. Among others, we prove that $$ \sum_{k=0}^{[n/4]}(-1)^k {n\choose 4k}\frac{B_{n-4k}(z) }{2^{6k}} =\frac{1}{2^{n+1}}\sum_{k=0}^{n} (-1)^k…
The discrete Chebyshev polynomials $t_n(x,N)$ are orthogonal with respect to a distribution, which is a step function with jumps one unit at the points $x=0,1,\cdots, N-1$, $N$ being a fixed positive integer. By using a double integral…
Our goal is to find an asymptotic behavior as $n\to\infty$ of the orthogonal polynomials $P_{n}(z)$ defined by Jacobi recurrence coefficients $a_{n}$ (off-diagonal terms) and $ b_{n}$ (diagonal terms). We consider the case $a_{n}\to\infty$,…
We enumerate bijectively the family of involutive Baxter permutations according to various parameters; in particular we obtain an elementary proof that the number of involutive Baxter permutations of size $2n$ with no fixed points is…
The Ap\'ery polynomials and in particular their asymptotic behavior play an essential role in the understanding of the irrationality of \zeta(3). In this paper, we present a method to study the asymptotic behavior of the sequence of the…
This paper begins with a re-examination of the Riemann-Siegel Integral, which first discovered amongst by Bessel-Hagen in 1926 and expanded upon by C. L. Siegel on his 1932 account of Riemanns unpublished work on the zeta function. By…
Already in 1734 Euler found a short explicit formula for the value of Riemann zeta function Zeta(s) when the argument s equals a positive integer 2n where n=1,2,3,. No such formula exists for odd positive integer arguments of Zeta. The…
We introduce a version of the asymptotic expansions for Bessel functions $J_\nu(z)$, $Y_\nu(z)$ that is valid whenever $|z| > \nu$ (which is deep in the Fresnel regime), as opposed to the standard expansions that are applicable only in the…
The modified Bessel function of the second kind $K_{i\nu}(x)$ of imaginary order for fixed $x>0$ possesses a countably infinite sequence of real zeros. Recently it has been shown that the $n$th zero behaves like $\nu_n\sim \pi n/\log\,n$ as…
The aim of this paper is to study the fully degenerate Bernoulli polynomials and numbers, which are a degenerate version of Bernoulli polynomials and numbers and arise naturally from the Volkenborn integral of the degenerate exponential…
We discuss the asymptotic expansions of certain products of Bernoulli numbers and factorials, e.g., \[ \prod_{\nu=1}^n |B_{2\nu}| \quad \text{and} \quad \prod_{\nu=1}^n (k \nu)!^{\nu^r} \quad \text{as} \quad n \to \infty \] for integers $k…
We find and discuss asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with recurrence coefficients $a_{n}, b_{n}$. Our main goal is to consider the case where off-diagonal elements $a_{n}\to\infty$ as $n\to\infty$. Formulas…
The (generalised) Mellin transforms of certain Chebyshev and Gegenbauer functions based upon the Chebyshev and Gegenbauer polynomials, have polynomial factors $p_n(s)$, whose zeros lie all on the `critical line' $\Re\,s=1/2$ or on the real…
The asymptotic behavior of the Mellin transform of the associated $B$-splines $B_N^*(t) :=t^{-N}B_N(t)$ with special knots in terms of theta-like functions is found. The proof is based on polynomial interpolation of power functions and…
The Eulerian number A(n,k) counts permutations of n symbols with exactly k descents. Motivated by problems in cryptography, several authors have studied the proportion of permutations whose number of descents lies in a fixed congruence…
We study the Borwein--Bailey--Girgensohn sinusoidal series S_BBG = sum_{n=1}^\infty (1/n) * ((2+sin n)/3)^n, originally posed as an open problem by Borwein, Bailey, and Girgensohn, whose convergence was established by Boppana using the…
Let $P(z)$ be a polynomial of degree $n$. In $2004$, Aziz and Rather \cite{aziz2004some} investigated the dependence of \[\bigg|P(Rz)-\alpha P(z)+\beta\biggl\{\biggl(\frac{R+1}{2}\biggr)^n-|\alpha|\biggr\}P(z)\bigg|, \ \text{for} \ z \in…
In the study of order estimation of the Riemann zeta-function $ \zeta(s) = \sum_{n=1}^\infty n^{-s} $, solving Lindel\"{o}f hypothesis is an important theme. As one of the relationships, asymptotic behavior of mean values has been studied.…
We study rather general multiple zeta-functions whose denominators are given by polynomials. The main aim is to prove explicit formulas for the values of those multiple zeta-functions at non-positive integer points. We first treat the case…