Related papers: Asymptotics and exact formulas for Zagier polynomi…

An urn contains black and red balls. Let $Z_n$ be the proportion of black balls at time $n$ and $0\leq L<U\leq 1$ random barriers. At each time $n$, a ball $b_n$ is drawn. If $b_n$ is black and $Z_{n-1}<U$, then $b_n$ is replaced together…

For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$…

In this paper, we introduce the polynomials $B^{(k)}_{n,\alpha}(x;q)$ generated by a function including Jackson $q$-Bessel functions $J^{(k)}_{\alpha}(x;q)$ $ (k=1,2,3),\,\alpha>-1$. The cases $\alpha=\pm\frac{1}{2}$ are the $q$-analogs of…

We compute asymptotic formulas for the $k^{\rm th}$ Fourier coefficients of $b_\lambda^n$, where $b_\lambda(z)=\frac{z-\lambda}{1-\lambda z}$ is the Blaschke factor associated to $\lambda\in\mathbb{D}$, $k\in[0,\infty)$ and $n$ is a large…

We study the asymptotic behavior of Multiple Meixner polynomials of first and second kind, respectively (see J. Arves\'u et al. J. Comput. Appl. Math., 153, (2003)). We use an algebraic function formulation for the solution of the…

The author derives new family of series representations for the values of the Riemann Zeta function $\zeta(s)$ at positive odd integers. For $n\in\mathbb{N}$, each of these series representing $\zeta(2n+1)$ converges remarkably rapidly with…

In his second notebook, Ramanujan discovered the following identity for the special values of $\zeta(s)$ at the odd positive integers \begin{equation*}\begin{aligned}\alpha^{-m}\,\left\{\dfrac{1}{2}\,\zeta(2m + 1) + \sum_{n =…

Building on our previous works on perturbative solutions to a Schwinger-Dyson for the massless Wess-Zumino model, we show how to compute 1/n corrections to its asymptotic behavior. The coefficients are analytically determined through a sum…

Strong asymptotics of polynomials orthogonal on the unit circle with respect to a weight of the form $$ W(z) = w(z) \prod_{k=1}^m |z-a_k|^{2\beta_k}, \quad |z|=1, \quad |a_k|=1, \quad \beta_k>-1/2, \quad k=1, ..., m, $$ where $w(z)>0$ for…

Motivated by a question in Schubert calculus, we study the interplay of quasisymmetric polynomials with the divided symmetrization operator, which was introduced by Postnikov in the context of volume polynomials of permutahedra. Divided…

We consider the asymptotic behavior of bounded solutions of the difference equations of the form $x(n+1)=Bx(n) + y(n)$ in a Banach space $\X$, where $n=1,2,...$, $B$ is a linear continuous operator in $\X$, and $(y(n))$ is a sequence in…

We study the signed Bernoulli convolution $$\nu_\beta^{(n)}=*_{j=1}^n \left (\frac12\delta_{\beta^{-j}}-\frac12\delta_{-\beta^{-j}}\right ),\ n\ge 1$$ where $\beta>1$ satisfies $$\beta^m=\beta^{m-1}+\cdots+\beta+1$$ for some integer $m\ge…

Let $f \in \mathbb{Z}[y]$ be a polynomial such that $f(\mathbb{N}) \subseteq \mathbb{N}$, and let $p_{\mathcal{A}_{f}}(n)$ denote number of partitions of $n$ whose parts lie in the set $\mathcal{A}_f:=\{f(n):n \in \mathbb{N}\}$. Under…

Using the expansion in a Fourier-Gegenbauer series, we prove several identities that extend and generalize known results. In particular, it is proved among other results, that \begin{equation*}…

We observe that five polynomial families have all of their zeros on the unit circle. We prove the statements explicitly for four of the polynomial families. The polynomials have coefficients which involve Bernoulli numbers, Euler numbers,…

A Fourier-type integral representation for Bessel's function of the first kind and complex order is obtained by using the Gegenbuaer extension of Poisson's integral representation for the Bessel function along with a trigonometric integral…

In 1848, Hermite introduced a reduction theory for binary forms of degree $n$ which was developed more fully in the seminal 1917 treatise of Julia. This canonical method of reduction made use of a new, fundamental, but irrational…

This paper is a study of power series, where the coefficients are binomial expressions (iterated finite differences). Our results can be used for series summation, for series transformation, or for asymptotic expansions involving Stirling…

The asymptotic expansion of digamma function is a starting point for the derivation of approximants for harmonic sums or Euler-Mascheroni constant. It is usual to derive such approximations as values of logarithmic function, which leads to…

We study the behavior of $r$-fold zeta-functions of Euler-Zagier type with identical arguments $\zeta_r(s,s,\ldots,s)$ on the real line. Our basic tool is an "infinite'' version of Newton's classical identities. We carry out numerical…