Three-Term Recurrence Relations for Confluent Basic Hypergeometric Series with Applications to q-Bessel Functions

TL;DR

This paper derives new three-term recurrence relations for basic hypergeometric series and q-Bessel functions, revealing explicit coefficients and their connections to q-Lommel polynomials, with applications to special functions.

Contribution

It introduces novel three-term recurrence relations for ${}_1 heta_1$ and ${}_0 heta_1$ series, extending previous work and applying to q-Bessel functions with explicit coefficient formulas.

Findings

Recurrence relations involve integer q-shifts of parameters and variables.

Coefficients are explicitly expressed as rational functions.

Relations include q-Lommel polynomials as special cases.

Abstract

We establish three-term recurrence relations for the ${}_1\phi_1$ and ${}_0\phi_1$ basic hypergeometric series involving multiplicative shifts of the parameters and the variable by integer powers of q. The coefficients of these recurrence relations are shown to be uniquely determined by the shift indices and are given explicitly in terms of rational functions. These recurrence relations arise as confluent limits of previously established recurrence relations for the ${}_2\phi_1$ basic hypergeometric series. As an application, we derive three-term recurrence relations for Jackson's second and third q-Bessel functions. These recurrence relations involve additive shifts in the order and multiplicative q-shifts in the variable, and their coefficients include the known q-Lommel polynomials as special cases.