Bilateral q-ultraspherical functions

2025/08/12 by Schlosser, Michael J.
#33D45 (Primary) 33D15 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.08908

Abstract

We introduce a bilateral extension of the continuous q-ultraspherical polynomials which we call bilateral q-ultraspherical functions. These functions are given as specific bilateral basic hypergeometric ₂ψ₂ series, they are analytic in a variable x=cosθ and depend on two parameters β and γ and on a base q. For these bilateral q-ultraspherical functions we derive a bilateral generating function, find a three-term recurrence relation, explain how they behave under the action of the Askey--Wilson divided difference operator, and show that they satisfy a type of shifted orthogonality.

Citations

• New proofs of theorems on q-orthogonal functions
• Some q-supercongruences from transformation formulas for basic\n hypergeometric series
• Koornwinder polynomials and affine Hecke algebras
• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• A right inverse of the Askey-Wilson operator
• The Associated Ultraspherical Polynomials and their q-Analogues

Related

• A second addition formula for continuous q-ultraspherical polynomials
• Asymptotic formulae of two divergent bilateral basic hypergeometric series
• A q-Exponential Operator Based on the Derivative of Order 1 and Summation of Bilateral Basic Hypergeometric Series
• On a generalization of the Rogers generating function
• On the Very-well-poised Bilateral Basic Hypergeometric ₅ψ₅ Series