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

Yuka Yamaguchi

Abstract

We establish three-term recurrence relations for the ${}_1φ_1$ and ${}_0φ_1$ 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 ${}_2φ_1$ 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.

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

Abstract

We establish three-term recurrence relations for the and 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 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.
Paper Structure (7 sections, 6 theorems, 117 equations)

This paper contains 7 sections, 6 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Explicit expressions for the polynomials
  3. Three-term recurrence relation for ${}_1\phi_1$
  4. An alternative recurrence relation for ${}_1\phi_1$
  5. Three-term recurrence relation for ${}_0\phi_1$
  6. Three-term recurrence relation for $J^{(3)}_\nu$
  7. Three-term recurrence relation for $J^{(2)}_\nu$

Key Result

Theorem 1

The pair $(S, T)$ satisfying $(3trr_1phi1)$ is uniquely determined by $(k,m,n)$. Consequently, $S$ and $T$ satisfy the relation Moreover, $S$ is given explicitly by where $P^{(1,1)}(k,m,n; a,c,x; q)$ is a polynomial in $x$ of degree at most whose explicit expression is given in P^(1,1).

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 6
  • Remark 7
  • Definition 8
  • Remark 9
  • Definition 10
  • ...and 7 more