Related papers: Generalized Umemura polynomials and Hirota-Miwa eq…

In this paper, we examine a generalized magma equation for rational values of two parameters, $m$ and $n$. Firstly, the similarity reductions are found using the Lie group method of infinitesimal transformations. The Painlev\'e ODE test is…

Induced by three gluons symmetry, Mandelstam variables $s$, $t$, $u$ symmetric expressions are widely involved in collider physics, especially in heavy quarkonium physics. In this work we study general form of $s$, $t$, $u$ symmetric…

Let $\mathbb{F}_q$ denote the finite field of characteristic $p$ and order $q$. Let $\mathbb{Z}_q$ denote the unramified extension of the $p$-adic rational integers $\mathbb{Z}_p$ with residue field $\mathbb{F}_q$. Given two positive…

We investigate semi-classical generalizations of the Charlier and Meixner polynomials, which are discrete orthogonal polynomials that satisfy three-term recurrence relations. It is shown that the coefficients in these recurrence relations…

We define some generalizations of the classical descent and inversion statistics on signed permutations that arise from the work of Sack and Ulfarsson [20] and called after width-k descents and width-k inversionsof type A in Davis's work…

In this paper we study the gap probability problem in the Gaussian Unitary Ensembles of $n$ by $n$ matrices : The probability that the interval $J := (-a,a)$ is free of eigenvalues. In the works of Tracy and Widom, Adler and Van Moerbeke…

Let $m,r\in\mathbb{Z}$ and $\omega\in\mathbb{R}$ satisfy $0\leqslant r\leqslant m$ and $\omega\geqslant1$. Our main result is a generalized continued fraction for an expression involving the partial binomial sum $s_m(r) =…

An integer of the form $p_m(x)= \frac{(m-2)x^2-(m-4)x}{2} \ (m\ge 3)$, for some integer $x$ is called a generalized polygonal number of order $m$. A ternary sum $\Phi_{i,j,k}^{a,b,c}(x,y,z)=ap_{i+2}(x)+bp_{j+2}(y)+cp_{k+2}(z)$ of…

We promote the recent research by Akiyama and Kaneko on the higher-order derivative values $\Phi_n^{(k)}(1)$ of the cyclotomic polynomials. This article focuses on Lehmer's explicit formula of $\Phi_n^{(k)}(1)/\Phi_n(1)$ as a polynomial of…

We formulate a general conjecture on the characteristic polynomials of S-decomposed T-ramified Iwasawa modules over the cyclotomic Z {\ell}-extension of a number field. We show that this conjecture is equivalent to the conjunctions of the…

We present the exact solution for a set of nonlinear algebraic equations $\frac{1}{z_l}= \pi d + \frac{2 d}{n} \sum_{m \neq l} \frac{1}{z_l-z_m}$. These were encountered by us in a recent study of the low energy spectrum of the Heisenberg…

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…

For each natural number $m\ge 3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. In this paper, with the help of the congruence theta function, we establish conditions on $a$, $b$,…

Universal enveloping algebras of braided m-Lie algebras and PBW theorem are obtained by means of combinatorics on words.

Let $\{q_n^{(\alpha,\beta,m)}(x)\}_{n\ge 0}$ be the orthonormal polynomials respect to the Sobolev-type inner product \begin{equation*} \langle f,g\rangle_{\alpha,\beta,m}=\sum_{k=0}^m \int_{-1}^{1}f^{(k)}(x)g^{(k)}(x)\,…

Let $K$ be a field. In this article, we derive a formula for the discriminant of a sequence $\{r_{A,n}+c r_{A,n-1}\}$ of polynomials. Here, $c \in K$ and $\{r_{A,n} \}$ is a sequence of polynomials satisfying a certain recurrence relation…

The even and odd Zernike Polynomials R_n^m(x) can be expanded into sums of even and odd Chebyshev Polynomials T_i(x). This manuscript provides closed forms for the rational expansion coefficients c_{n,m,i} for a set of small 0 <= n-m <= 6…

We consider roots of a generalized Lens polynomial $L(z,\bar z)={\bar z}^m q(z)-p(z)$ and also harmonically splitting Lens type polynomial $L^{hs}(z,\bar z)=r(\bar z)q(z)-p(z)$ and with ${\rm deg}\,q(z)=n$, ${\rm deg}\,r(\bar z)=m$ and…

We survey some of the major results about normal Hilbert polynomials of ideals. We discuss a formula due to Lipman for complete ideals in regular local rings of dimension two, theorems of Huneke, Itoh, Huckaba, Marley and Rees in…

The aim of the paper is to extend the class of generalized Weyl algebras to a larger class of rings (they are also called {\em generalized Weyl algebras}) that are determined by two ring endomorphisms rather than one as in the case of `old'…