[Paper Review] A Survey on q-Polynomials and their Orthogonality Properties

Roberto S. Costas-Santos, Joaquín Sánchez-Lara|arXiv (Cornell University)|Feb 24, 2010

Mathematical functions and polynomials25 references3 citations

TL;DR

This paper presents a comprehensive survey on q-orthogonal polynomials, focusing on the orthogonality properties of Askey-Wilson and big q-Jacobi polynomials for complex parameters and |q| ≠ 1. It extends Favard’s theorem to degenerate cases where γ_N = 0, establishing Sobolev-type orthogonality and deriving factorizations of the form p_{n+N} = p_N p_n^{(N)}, linking families across the q-Askey and Nikiforov-Uvarov tableaux.

ABSTRACT

In this paper we study the orthogonality conditions satisfied by the classical q-orthogonal polynomials that are located at the top of the q-Hahn tableau (big q-jacobi polynomials (bqJ)) and the Nikiforov-Uvarov tableau (Askey-Wilson polynomials (AW)) for almost any complex value of the parameters and for all non-negative integers degrees. We state the degenerate version of Favard's theorem, which is one of the keys of the paper, that allow us to extend the orthogonality properties valid up to some integer degree N to Sobolev type orthogonality properties. We also present, following an analogous process that applied in [16], tables with the factorization and the discrete Sobolev-type orthogonality property for those families which satisfy a finite orthogonality property, i.e. it consists in sum of finite number of masspoints, such as q-Racah (qR), q-Hahn (qH), dual q-Hahn (dqH), and q-Krawtchouk polynomials (qK), among others. -- [16] R. S. Costas-Santos and J. F. Sanchez-Lara. Extensions of discrete classical orthogonal polynomials beyond the orthogonality. J. Comp. Appl. Math., 225(2) (2009), 440-451

Motivation & Objective

• To investigate the orthogonality conditions satisfied by classical q-orthogonal polynomials, particularly Askey-Wilson and big q-Jacobi polynomials, for almost any complex parameter values and all non-negative integer degrees.
• To extend Favard’s theorem to degenerate cases where the recurrence coefficient γ_N vanishes, enabling the characterization of Sobolev-type orthogonality for finite-degree polynomial sequences.
• To systematically derive factorization identities of the form p_{n+N} = p_N p_n^{(N)} for q-polynomials with finite orthogonality (e.g., q-Racah, q-Hahn, dual q-Hahn, q-Krawtchouk), linking them to associated families in the q-Askey and Nikiforov-Uvarov tableaux.
• To compile and present tables detailing the discrete Sobolev-type orthogonality and factorization structures for q-polynomials with finite mass-point orthogonality, providing a unified reference for these families.

Proposed method

• Utilizes the second-order hypergeometric-type difference operator H defined via forward and backward difference operators, with σ̂(x(s)) and τ(x(s)) as polynomials of degree ≤2 and 1, respectively.
• Applies the degenerate version of Favard’s theorem, which characterizes orthogonal polynomials via a three-term recurrence relation (TTRR) even when γ_N = 0 for some N, leading to Sobolev-type orthogonality.
• Derives the factorization p_{n+N} = p_N p_n^{(N)} where p_n^{(N)} is the Nth associated polynomial, using the structure of the recurrence coefficients and the action of the difference operator T = Δ/Δx(s).
• Establishes unnormalized identities between q-polynomials (e.g., q-Racah to Askey-Wilson, big q-Jacobi to q-Hahn) via parameter transformations and shift operators, using known identities from the literature.
• Constructs tables mapping each q-polynomial family (e.g., q-Hahn, dual q-Hahn, q-Krawtchouk) to its associated polynomial and the corresponding T^N(p_{n+N}) expression, enabling orthogonality characterization.
• Analyzes the case |q| = 1 separately, noting that the orthogonality structure remains valid under this condition, though convergence and measure properties may differ.

Experimental results

Research questions

• RQ1 How do the orthogonality properties of Askey-Wilson and big q-Jacobi polynomials behave for almost any complex parameter values and all non-negative integer degrees?
• RQ2 What is the structure of the Sobolev-type orthogonality that arises when the recurrence coefficient γ_N vanishes in the three-term recurrence relation?
• RQ3 Can the factorization p_{n+N} = p_N p_n^{(N)} be systematically derived for q-polynomials with finite orthogonality (i.e., finite number of mass points), and what is the associated polynomial p_n^{(N)}?
• RQ4 How are the various q-polynomial families (e.g., q-Racah, q-Hahn, q-Krawtchouk) related through unnormalized identities involving shift operators and parameter transformations?
• RQ5 What is the role of the difference operator T = Δ/Δx(s) in generating the associated polynomial sequences and linking families across the q-Askey and Nikiforov-Uvarov tableaux?

Key findings

• For Askey-Wilson and big q-Jacobi polynomials, the orthogonality conditions are valid for all non-negative integer degrees and almost any complex parameter values, including |q| ≠ 1.
• When γ_N = 0 in the TTRR, the degenerate Favard’s theorem implies that the polynomial sequence satisfies a Sobolev-type orthogonality with respect to a linear functional involving the Nth iterate of the difference operator T.
• The factorization p_{n+N} = p_N p_n^{(N)} holds for q-polynomials with finite orthogonality, where p_n^{(N)} is an associated polynomial belonging to the same or a related family in the q-Askey or Nikiforov-Uvarov tableaux.
• For q-Racah polynomials, setting α = q^{-N} causes γ_N = 0, and the associated polynomial r_n^{(N)} is explicitly given in terms of Askey-Wilson polynomials with transformed parameters.
• The operator T^N(p_{n+N}) is shown to be proportional to a polynomial from a higher-level family (e.g., Askey-Wilson or big q-Jacobi), with parameters scaled by q^N, confirming the structure of the Sobolev orthogonality.
• Tables are provided that map each q-polynomial family (e.g., q-Hahn, dual q-Hahn, q-Krawtchouk) to its associated polynomial and the corresponding T^N(p_{n+N}) expression, enabling systematic identification of orthogonality and factorization structures.