Duality of q-polynomials, orthogonal on countable sets of points

Abstract

We review properties of q-orthogonal polynomials, related to their orthogonality, duality and connection with the theory of symmetric (self-adjoint) operators, represented by a Jacobi matrix. In particular, we show how one can naturally interpret the duality of families of q-polynomials, orthogonal on countable sets of points. In order to obtain orthogonality relations for dual sets of polynomials, it is proposed to use two symmetric (self-adjoint) operators, representable (in some distinct bases) by Jacobi matrices. This approach is applied to several pairs of dual families of q-polynomials, orthogonal on countable sets, from the q-Askey scheme. For each such pair, the corresponding operators, representable by Jacobi matrices, are explicitly given. These operators are employed in order to find explicitly sets, on which the polynomials are orthogonal, and orthogonality relations for them.

71 pages, corrected typos, added references

References in corpus (4)

• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an algebraic approach to the Askey scheme of orthogonal polynomials
• Big q-Laguerre and q-Meixner polynomials and representations of the algebra U_q(su(1,1))
• On discrete q-ultraspherical polynomials and their duals
• Bilinear summation formulas from quantum algebra representations

Cited by in corpus (11)

• Orthogonal Polynomials from Hermitian Matrices
• Orthogonal Polynomials from Hermitian Matrices II
• Dualities in the $q$-Askey scheme and degenerate DAHA
• Limits of elliptic hypergeometric biorthogonal functions
• Exactly solvable discrete time Birth and Death processes
• Charting the $q$-Askey scheme. II. The $q$-Zhedanov scheme
• Scaling Limits for the Gibbs States on Distance-Regular Graphs with Classical Parameters
• On the shape of a tridiagonal pair
• Representations of the Quantum Algebra su_q(1,1) and Discrete q-Ultraspherical Polynomials
• On Orthogonality Relations for Dual Discrete q-Ultraspherical Polynomials
• Two non-nilpotent linear transformations that satisfy the cubic $q$-Serre relations