Related papers: A Survey on q-Polynomials and their Orthogonality …
The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the P\"oschl-Teller potential (Odake-Sasaki). By fine-tuning the parameter(s) of…
We consider a one-parameter family of nonlinear coherent states by replacing the factorial in coefficients of the canonical coherent states by a specific generalized factorial depending on a parameter gamma. These states are superposition…
It is well-known that orthogonal polynomials on the real line satisfy a three-term recurrence relation and conversely every system of polynomials satisfying a three-term recurrence relation is orthogonal with respect to some positive Borel…
In this contribution we consider sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product \[ \langle f,g \rangle _{S}:= \langle {\bf u}, f g\rangle +N (\mathscr D_q f)(\alpha) (\mathscr D _{q}g)(\alpha),\qquad…
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 define a one-parameter family of two-sided coideals in U_q(gl(n)) and study the corresponding algebras of infinitesimally right invariant functions on the quantum unitary group U_q(n). The Plancherel decomposition of these algebras with…
We construct new examples of exceptional Hahn and Jacobi polynomials. Exceptional polynomials are orthogonal polynomials with respect to a measure which are also eigenfunctions of a second order difference or differential operator. The most…
We expand the Chebyshev polynomials and some of its linear combination in linear combinations of the q-Hermite, the Rogers (q-utraspherical) and the Al-Salam--Chihara polynomials and vice versa. We use these expansions to obtain expansions…
A limit formula from q-Racah polynomials to big q-Jacobi polynomials is given which can be considered as a limit formula for orthogonal polynomials. This is extended to a multi-parameter limit with 3 parameters, also involving (q-)Hahn…
In this paper we give a characterization of some classical q-orthogonal polynomials in terms of a difference property of the associated Stieltjes function, i.e this function solves a first order non-homogeneous q-difference equation. The…
This paper is to evaluate certain Catalan-Hankel Pfaffians by the theory of skew orthogonal polynomials. Due to different kinds of hypergeometric orthogonal polynomials underlying the Askey scheme, we explicitly construct the classical skew…
We develop a theory of Jacobi polynomials for parabolic subgroups of finite reflection groups that specializes to the cases studied by Heckman and Opdam in which the whole group and the trivial group are considered. For the intermediate…
Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line…
Properties of the $q$-ultraspherical polynomials for $q$ being a primitive root of unity are derived using a formalism of the $so_q(3)$ algebra. The orthogonality condition for these polynomials provides a new class of trigonometric…
In this paper, we study the sequence of orthogonal polynomials $\{S_n\}_{n=0}^{\infty}$ with respect to the Sobolev-type inner product $$\langle f,g \rangle= \int_{-1}^{1} f(x) g(x) \,d\mu(x) +\sum_{j=1}^{N} \eta_{j} \,f^{(d_j)}(c_{j})…
In this paper, we show that the classical discrete orthogonal univariate polynomials (namely, Hahn polynomials on an equidistant lattice with unit weights) of sufficiently high degrees have extremely small values near the endpoints (we call…
From analysis of a big variety of different knots we conclude that at q which is an root of unity, q^{2m}=1, HOMFLY polynomials in symmetric representations [r] satisfy recursion identity: H_{r+m} = H_r H_m for any A, which is a…
This is a review of ($q$-)hypergeometric orthogonal polynomials and their relation to representation theory of quantum groups, to matrix models, to integrable theory, and to knot theory. We discuss both continuous and discrete orthogonal…
New bispectral orthogonal polynomials are obtained from an unconventional truncation of the Askey-Wilson polynomials. In the limit $q \to 1$, they reduce to the para-Racah polynomials which are orthogonal with respect to a quadratic…
By using a generalization of Sturm-Liouville problems in $q$-difference spaces, a class of symmetric $q$-orthogonal polynomials with four free parameters is introduced. The standard properties of these polynomials, such as a second order…