Related papers: Jacobi-Stirling polynomials and $P$-partitions

In this note we deduce well known modular identities for Jacobi theta functions using the spectral representations associated with the real valued Brownian motion taking values on $[-1,+1]$. We consider two cases: (i) reflection at $-1$ and…

In this paper, we study a family of single variable integral representations for some products of $\zeta(2n+1)$, where $\zeta(z)$ is Riemann zeta function and $n$ is positive integer. Such representation involves the integral…

We study sums of the form $\sum_{k=m}^n a_{nk} b_{km}$, where $a_{nk}$ and $b_{km}$ are binomial coefficients or unsigned Stirling numbers. In a few cases they can be written in closed form. Failing that, the sums still share many common…

Integrals involving derivatives of Legendre polynomials frequently arise in applications ranging from multipole expansions for processes involving electromagnetic probes to spectral methods in numerical physics. Despite their practical…

The Stirling numbers of the first kind can be represented in terms of a new class of polynomials that are closely related to the Bernoulli polynomials. Recursion relations for these polynomials are given.

Let $\alpha$ and $\beta$ be two nonnegative integers such that $\beta < \alpha$. For an arbitrary sequence $\{a_n\}_{n\geqslant 1}$ of complex numbers, we consider the generalized Lambert series in order to investigate linear combinations…

Using $p$-adic numbers, we partially categorize the cycles of a sizable class of polynomial dynamical systems. In turn, we prove a few results related to the non-trivial cycles of the $\textit{Collatz map}$ $\text{Col} : \mathbb{Z}_+ \to…

Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n,$ where the coefficients $a_j,$ $j \in \{0,1,2,\cdots n\},$ are real numbers. We impose some restriction on the coefficients and then prove some extensions and…

We make an application of ideas from partition theory to a problem in multiplicative number theory. We propose a deterministic model of prime number distribution, from first principles related to properties of integer partitions, that…

We study the average number of representations of an integer $n$ as $n = \phi(n_{1}) + \dots + \phi(n_{j})$, for polynomials $\phi \in \mathbb{Z}[n]$ with $\partial\phi = k\ge 1$, $\operatorname{lead}(\phi) = 1$, $j \ge k$, where $n_{i}$ is…

Given a prime $p$ and a positive integer $k$, let $\mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ be the ring of $n \times n$ matrices over $\mathbb{Z}/p^{k}\mathbb{Z}$. We consider the number of solutions $X \in…

The author advocates two specific mathematical notations from his popular course and joint textbook, "Concrete Mathematics". The first of these, extending an idea of Iverson, is the notation "[P]" for the function which is 1 when the…

Raimi's theorem guarantees the existence of a partition of $\mathbb{N}$ into two parts with an unavoidable intersection property: for any finite coloring of $\mathbb{N}$, some color class intersects both parts infinitely many times, after…

This paper investigates coefficients of cyclotomic polynomials theoretically and experimentally. We prove the following result. {{\em If $n=p_1\ldots p_k$ where $p_i$ are odd primes and $p_1<p_2<\ldots<p_r<p_1+p_2<p_{r+1}<\ldots<p_t$ with…

Representation of analytic functions as convergent series in Jacobi polynomials $P_n^{(a,b)}$ is reformulated using a unified approach for almost all complex $a, b$. The coefficients of the series are given as usual integrals in the…

In this paper, we present the compositional inverses of several classes permutation polynomials of the form $\sum_{i=1}^kb_i(x^{p^m}+x+\delta)^{s_i}-x$ over $\mathbb{F}_{p^{2m}}$, where for $1\leq i \leq k,$ $s_i, m$ are positive integers,…

Let $p$ be an odd prime. In this paper, we study the permutation behaviour of the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ when $n=p^{l_1}+3$, $n=p^{l_1}+p^{l_2}+p^{l_3}$, and $n=p^{l_1}+p^{l_2}+p^{l_3}+p^{l_4}$,…

The generating function of the Bannai-Ito polynomials is derived using the fact that these polynomials are known to be essentially the Racah or $6j$ coefficients of the $\mathfrak{osp}(1|2)$ Lie superalgebra. The derivation is carried in a…

Using properties of Gauss and Jacobi sums, we derive explicit formulas for the number of solutions to a diagonal equation of the form $x_1^{2^m}+\dots+x_n^{2^m}=0$ over a finite field of characteristic $p\equiv\pm 3\pmod{8}$. All of the…

Using the ground state $\psi_0$ of a multicomponent generalization of the Calogero-Sutherland model as a weight function, orthogonal polynomials in the coordinates of one of the species are constructed. Using evidence from exact analytic…