Related papers: Characterization of Orthogonal Polynomials on latt…

Let $A$ be a $r\times r$ rational nonzero symmetric matrix, $B$ a rational column vector, $C$ a rational scalar. For any integer lattice $L$ and vector $v$ of $\mathbb{Z}^r$, we define Nahm sum on the lattice coset $v+L\in \mathbb{Z}^r/L$:…

This paper addresses the factorization of polynomials of the form $F(x) = f_{0}(x) + f_{1}(x) x^{n} + \cdots + f_{r-1}(x) x^{(r-1)n} + f_{r}(x) x^{rn}$ where $r$ is a fixed positive integer and the $f_{j}(x)$ are fixed polynomials in…

Consider a semi-infinite skew-symmetric moment matrix, $m_{\iy}$ evolving according to the vector fields $\pl m / \pl t_k=\Lb^k m+m \Lb^{\top k} ,$ where $\Lb$ is the shift matrix. Then the skew-Borel decomposition $ m_{\iy}:= Q^{-1} J…

In this paper we consider the characteristic polynomial of not necessarily ranked posets. We do so by allowing the rank to be an arbitrary function from the poset to the nonnegative integers. We will prove two results showing that the…

We derive semiclassical asymptotics for the orthogonal polynomials P_n(z) on the line with respect to the exponential weight \exp(-NV(z)), where V(z) is a double-well quartic polynomial, in the limit when n, N \to \infty. We assume that…

Let $z_1, \dots, z_m$ be $m$ distinct complex numbers, normalized to $|z_k| = 1$, and consider the polynomial $$ p_{m}(z) = \prod_{k=1}^{m}{(z-z_k)}.$$ We define a sequence of polynomials in a greedy fashion, $$ p_{N+1}(z) = p_{N}(z)…

In this paper we present a complete method for finding the roots of all polynomials of the form $\phi(z)=c_n z^n+c_{n-1} z^{n-1}+\dots+c_1 z+c_0$ over a given octonion division algebra. When $\phi(z)$ is monic we also consider the companion…

A class P_{n,m,p}(x) of polynomials is defined. The combinatorial meaning of its coefficients is given. Chebyshev polynomials are the special cases of P_{n,m,p}(x). It is first shown that P_{n,m,p}(x) may be expressed in terms of…

In this paper we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, A_n(z) and B_n(z), appearing in the ladder operators satisfy the two…

In his monograph [Classical and quantum orthogonal polynomials in one variable, Cambridge University Press, 2005 (paperback edition 2009)], Ismail conjectured that certain structure relations involving the Askey-Wilson operator characterize…

In this paper, we study polynomial-like elements in vector spaces equipped with group actions. We first define these elements via iterated difference operators. In the case of a full rank lattice acting on an Euclidean space, these…

In this paper, we exhibit explicitly a sequence of $2\times2$ matrix valued orthogonal polynomials with respect to a weight $W_{p,n}$, for any pair of real numbers $p$ and $n$ such that $0<p<n$. The entries of these polynomiales are…

We study the set $\mathcal{L}_{F}$ of all $F$-vector spaces $L(P)$ where $P$ is monic and splits over $F$ and $L(Q)$ denotes the set of linear recurrence sequences over $F$ with characteristic polynomial $Q$. We show that $\mathcal{L}_{F}$…

We study the class of Lorentzian polynomials. The class contains homogeneous stable polynomials as well as volume polynomials of convex bodies and projective varieties. We prove that the Hessian of a nonzero Lorentzian polynomial has…

In this paper, a fourth-order partial divided-difference equation on quadratic lattices with polynomial coefficients satisfied by bivariate Racah polynomials is presented. From this equation we obtain explicitly the matrix coefficients…

Let $(P_N)_{N\ge0}$ one of the classical sequences of orthogonal polynomials, i.e., Hermite, Laguerre or Jacobi polynomials. For the roots $z_{1,N},\ldots, z_{N,N}$ of $P_N$ we derive lower estimates for $\min_{i\ne j}|z_{i,N}-z_{j,N}|$ and…

We present a method to obtain weight functions associated with linear and quadratic lattices that yield discrete orthogonality with respect to a quasi-definite moment functional of the orthogonal polynomial sequences in the Askey scheme,…

$q$-Analogues of the coefficients of $x^a$ in the expansion of $\prod_{j=1}^N (1+x+...+x^j)^{L_j}$ are proposed. Useful properties, such as recursion relations, symmetries and limiting theorems of the ``$q$-supernomial coefficients'' are…

Two families of d-orthogonal polynomials related to su(2) are identified and studied. The algebraic setting allows their full characterization (explicit expressions, recurrence relations, difference equations, generating functions, etc.) of…

$2\times2$ matrix polynomials of the form $P_{n}(z)= \Sigma^{n}_{j=0}\,\sigma_{j}\,z^{j}$, for the cases $n=1,2,3$ are constructed, and the nature of PT-symmetry is examined across different points $z=(x,y)$ in the complex plane. The…