Related papers: A Survey on q-Polynomials and their Orthogonality …

Orthogonality of the Jacobi and of Laguerre polynomials, P_n^(a,b) and L_n^(a), is established for a,b complex (a,b not negative integers and a+b different from -2,-3,...) using the Hadamard finite part of the integral which gives their…

The five parameter family of multivariable Askey-Wilson polynomials is studied with four parameters generically complex. The multivariable Askey-Wilson polynomials form an orthogonal system with respect to an explicit (in general complex)…

We prove that for |x|,|t|<1, -1 <q \leq1 and n\geq0: \Sigma_{i\geq0}((t^{i})/((q)_{i}))h_{n+i}(x|q) = h_{n}(x|t,q) \Sigma_{i\geq0}((t^{i})/((q)_{i}))h_{i}(x|q), where h_{n}(x|q) and h_{n}(x|t,q) are respectively the so called q-Hermite and…

We introduce the -1 dual Hahn polynomials through an appropriate $q \to -1$ limit of the dual q-Hahn polynomials. These polynomials are orthogonal on a finite set of discrete points on the real axis, but in contrast to the classical…

A high order linear $q$-difference equation with polynomial coefficients having $q$-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation is related to the number of orthogonality conditions that these…

An orthomorphism over a finite field $\mathbb{F}_q$ is a permutation $\theta:\mathbb{F}_q\mapsto\mathbb{F}_q$ such that the map $x\mapsto\theta(x)-x$ is also a permutation of $\mathbb{F}_q$. The degree of an orthomorphism of $\mathbb{F}_q$,…

We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part of the paper we prove a number properties…

We study two families of orthogonal polynomials. The first is a finite family related to the Askey-Wilson polynomials but the orthogonality is on the real line. A limiting case of this family is an infinite system of orthogonal polynomials…

We state and prove the $q$-extension of a result due to Johnston and Jordaan (cf. \cite{Johnston-2015}) and make use of this result, the orthogonality of $q$-Laguerre, little $q$-Jacobi, $q$-Meixner and Al-Salam-Carlitz I polynomials as…

We introduce a new family of special functions, namely $q$-Charlier multiple orthogonal polynomials. These polynomials are orthogonal with respect to $q$-analogues of Poisson distributions. We focus our attention on their structural…

We study the relation between certain non-degenerate lower Hessenberg infinite matrices $\mathcal{G}$ and the existence of sequences of orthogonal polynomials with respect to Sobolev inner products. In other words, we extend the well-known…

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…

A hyperoval in the projective plane $\mathbb{P}^2(\mathbb{F}_q)$ is a set of $q+2$ points no three of which are collinear. Hyperovals have been studied extensively since the 1950s with the ultimate goal of establishing a complete…

We investigate polynomials, called m-polynomials, whose generator polynomial has coefficients that can be arranged as a matrix, where q is a positive integer greater than one. Orthogonality relations are established and coefficients are…

The q-classical orthogonal polynomials of the q-Hahn Tableau are characterized from their orthogonality condition and by a first and a second structure relation. Unfortunately, for the q-semiclassical orthogonal polynomials (a…

The study of $P$-polynomial association schemes (distance-regular graphs) and $Q$-polynomial association schemes, and in particular $P$- and $Q$-polynomial association schemes, has been a central theme not only in the theory of association…

We review the properties of six families of orthogonal polynomials that form the main bulk of the collection called the Askey--Wilson scheme of polynomials. We give connection coefficients between them as well as the so-called linearization…

We study properties of compactly supported, 4 parameter \newline $(\rho _{12},\rho _{23},\rho _{13},q)\in (-1,1)^{\times 4}$ family of continuous type 3 dimensional distributions, that have the property that for $q\rightarrow 1^{-}$ this…

In this paper, we establish a $q$-integral formula by using the orthogonality relation, and also provide a new proof of the $q$-orthogonality relation for the continuous $q$-ultraspherical polynomials. A new $q$-beta integral with five…

We prove new combinatorial results about polynomial configurations in large subsets of finite fields. Bergelson--Leibman--McCutcheon (2005) showed that for any polynomial $P(x) \in \mathbb{Z}[x]$ with $P(0) = 0$, if $A \subseteq…