Related papers: A limit $q=-1$ for the big q-Jacobi polynomials
We offer a detailed treatment of spectral and Weyl-Titchmarsh-Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression \begin{align*} \tau_{\alpha,\beta} = - (1-x)^{-\alpha} (1+x)^{-\beta}(d/dx)…
We show how to obtain linear combinations of polynomials in an orthogonal sequence $\{P_n\}_{n\geq 0}$, such as $Q_{n,k}(x)=\sum\limits_{i=0}^k a_{n,i}P_{n-i}(x)$, $a_{n,0}a_{n,k}\neq0$, that characterize quasi-orthogonal polynomials of…
We study a family of Jacobi operators in which the diagonal entries are multiplied by a coupling parameter $\lambda\geq0$. Under suitable conditions, the operator is self-adjoint for every $\lambda>0$, while the formal limit at $\lambda=0$…
We discuss quadratic transformations for orthogonal polynomials in one and two variables. In the one-variable case we list many (or all) quadratic transformations between families in the Askey scheme or $q$-Askey scheme. In the two-variable…
The spectral properties of two special classes of Jacobi operators are studied. For the first class represented by the $2M$-dimensional real Jacobi matrices whose entries are symmetric with respect to the secondary diagonal, a new…
Functions like the exponential, Chebyshev polynomials, and monomial symmetric polynomials are preeminent among all special functions. They have simple definitions and can be expressed using easily specified integers like n!. Families of…
Three $q$-versions of Lommel polynomials are studied. Included are explicit representations, recurrences, continued fractions, and connections to associated Askey--Wilson polynomials. Combinatorial results are emphasized, including a…
Let ${\bf P}_k^{(\alpha, \beta)} (x)$ be an orthonormal Jacobi polynomial of degree $k.$ We will establish the following inequality \begin{equation*} \max_{x \in [\delta_{-1},\delta_1]}\sqrt{(x- \delta_{-1})(\delta_1-x)}…
Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical…
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…
In this paper, we introduce the notion of the complete joint Jacobi polynomial of two linear codes of length $n$ over $\mathbb{F}_q$ and $\mathbb{Z}_k$. We give the MacWilliams type identity for the complete joint Jacobi polynomials of…
We investigate type I multiple orthogonal polynomials on $r$ intervals which have a common point at the origin and endpoints at the $r$ roots of unity $\omega^j$, $j=0,1,\ldots,r-1$, with $\omega = \exp(2\pi i/r)$. We use the weight…
A model of the Bannai-Ito algebra in a superspace is introduced. It is obtained from the three-fold tensor product of the basic realization of the Lie superalgebra $osp(1|2)$ in terms of operators in one continuous and one Grassmanian…
Computing asymptotics of the recurrence coefficients of X1-Jacobi polynomials we investigate the limit of Christoffel function. We also study the relation between the normalized counting measure based on the zeros of the modified average…
In this paper the general Krall-type Askey-Wilson polynomials are introduced. These polynomials are obtained from the Askey-Wilson polynomials via the addition of two mass points to the weight function of them at the points $\pm1$. Several…
Ordinary orthogonal polynomials are uniquely characterized by the three term recurrence relations up to an overall multiplicative constant. We show that the newly discovered M-indexed orthogonal polynomials satisfy 3+2M term recurrence…
Following Verde-Star, Linear Algebra Appl. 627 (2021), we label families of orthogonal polynomials in the $q=1$ Askey scheme together with their hypergeometric representations by three sequences $x_k, h_k, g_k$ of polynomials in $k$, two of…
Studying the isotropy orbits of compact symmetric spaces Reiswich introduced a family of explicit polynomials in one variable in order to describe the unique minimal isotropy orbit of compact symmetric spaces with Dynkin diagram of type…
In the present paper, we provide results that relate the Jacobi polynomials in genus $g$. We show that if a code is $t$-homogeneous that is, the codewords of the code for every given weight hold a $t$-design, then its Jacobi polynomial in…
In this paper we introduce and discuss some classes of orthogonal polynomials in several non-commuting variables. The emphasis is on a non-commutative version of the orthogonal polynomials on the real line. We introduce recurrence equations…