Related papers: On another characterization of Askey-Wilson polyno…

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)…

Let $(x_n)_{n\geq0}$ be a linear recurrence sequence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In 2017,…

Given a sequence of orthogonal polynomials $(p_n)_n$ with respect to a positive measure in the real line, we study the real zeros of finite combinations of $K+1$ consecutive orthogonal polynomials of the form $$…

We consider the exactly solvable quantum mechanical systems whose eigenfunctions are described by the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. Corresponding to the recurrence relations with…

We explore differential operators, $T$, that diagonalize on a simple basis, $\{B_n(x)\}_{n=0}^\infty$, with respect to some sequence of real numbers, $\{a_n\}_{n=0}^\infty$, and sequence of polynomials, $\{Q_k(x)\}_{k=0}^\infty$, as in $…

We study a limiting case of the Askey-Wilson polynomials when one of the parameters goes to infinity, namely continuous dual q-Hahn polynomials when q > 1. Solutions to the associated indeterminate moment problem by general theory are found…

Orthogonal polynomials with respect to the hypergeometric distribution on lattices in polyhedral domains in ${\mathbb R}^d$, which include hexagons in ${\mathbb R}^2$ and truncated tetrahedrons in ${\mathbb R}^3$, are defined and studied.…

The discrete orthogonality relations for the multi-indexed orthogonal polynomials in discrete quantum mechanics with pure imaginary shifts are investigated. We show that the discrete orthogonality relations hold for the case-(1)…

We can write the polynomial solution of the second order linear differential equation of hypergeometric-type $$ \phi(x)y''+\psi(x)y'+\lambda y=0, $$ where $\phi$ and $\psi$ are polynomials, $\deg \phi\le 2$, $\deg \psi=1$ and $\lambda$ is a…

We find and discuss asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with recurrence coefficients $a_{n}, b_{n}$. Our main goal is to consider the case where off-diagonal elements $a_{n}\to\infty$ as $n\to\infty$. Formulas…

In this survey we summarize the current state of known orthogonality relations for the $q$ and $q^{-1}$-symmetric and dual subfamilies of the Askey--Wilson polynomials in the $q$-Askey scheme. These polynomials are the continuous dual $q$…

An explicit structure relation for Askey-Wilson polynomials is given. This involves a divided q-difference operator which is skew symmetric with respect to the Askey-Wilson inner product and which sends polynomials of degree n to…

We investigate the quadratic decomposition and duality to classify symmetrical $H_{q}$-semiclassical orthogonal $q$-polynomials of class one where $H_{q}$ is the Hahn's operator. For any canonical situation, the recurrence coefficients, the…

We show that any differential operator of the form $L(y)=\sum_{k=0}^{k=N} a_{k}(x) y^{(k)}$, where $a_k$ is a real polynomial of degree $\leq k$, has all real eigenvalues in the space of polynomials of degree at most n, for all n. The…

In this paper, we will discuss the notion of almost orthogonality in a functional sequence.Especially, we will define a few sequences of almost orthogonal polynomials which can be used successfully for modeling of electronic systems which…

We study a sequence of polynomials orthogonal with respect to a one parameter family of weights $$ w(x):=w(x,t)=\rex^{-t/x}\:x^{\al}(1-x)^{\bt},\quad t\geq 0, $$ defined for $x\in[0,1].$ If $t=0,$ this reduces to a shifted Jacobi weight.…

For the wavelet type orthonormal systems $\phi_n$, we establish a new bound \begin{equation} \left\|\max_{1\le m\le n}\left|\sum_{j\in G_m}\langle f,\phi_j\rangle \phi_j\right|\right\|_p\lesssim \sqrt{\log (n+1)}\cdot \|f\|_p,\quad…

Four families of generalizations of trigonometric functions were recently introduced. In the paper the functions are transformed into four families of orthogonal polynomials depending on two variables. Recurrence relations for construction…

The problem of finding measures whose orthogonal polynomials are also eigenfunctions of higher-order difference operators have been recently solved by multiplying the classical discrete measures by suitable polynomials. This problem was…

Eigenfunctions of the Askey-Wilson second order $q$-difference operator for $0<q<1$ and $|q|=1$ are constructed as formal matrix coefficients of the principal series representation of the quantized universal enveloping algebra…