Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The $q$-Laplace Transforms compared: the basic confluent hypergeometric function ${}_2φ_0$

Daniel Meikle, Adri Olde Daalhuis

TL;DR

The paper develops and compares three $q$-Borel–Laplace resummations for divergent $q$-Gevrey series arising from $q$-difference equations, revealing that their differences are exactly $q$-exponentially small and governed by universal Stokes-type structures. It then specializes to the simplest divergent basic hypergeometric function ${}_2\phi_0$, introducing three resummed versions ${\rm U}_{q,e}(a,b;z)$ and deriving a comprehensive toolkit: integral, sum, Mellin–Barnes representations, and explicit connection formulas to the standard $q$-Kummer solutions, together with sharp error bounds. A key result is that the ${\rm E}$-version, via Mellin–Barnes representations, is the most natural resummation, while the discrete ${\rm \lambda}$-version offers a one-valued alternative with a family of poles. The work also yields a discrete orthogonality for Stieltjes–Wigert polynomials and highlights the interplay between three $q$-Laplace transforms in driving $q$-special function theory.

Abstract

In solving $q$-difference equations, and in the definition of $q$-special functions, we encounter formal power series in which the $n$th coefficient is of size $q^{-\binom{n}{2}}$ with $q\in(0,1)$ fixed. To make sense of these formal series, a $q$-Borel-Laplace resummation is required. There are three candidates for the $q$-Laplace transform, resulting in three different resummations. Surprisingly, the differences between these resummations have not been discussed in the literature. Our main result provides explicit formulas for these $q$-exponentially small differences. We also give simple Mellin--Barnes integral representations for all the basic hypergeometric ${}_rφ_s$ functions and derive a third (discrete) orthogonality condition for the Stieltjes--Wigert polynomials. As the main application, we introduce three resummations for the ${}_2φ_0$ functions which can be seen as $q$ versions of the Kummer $U$ functions. We derive many of their properties, including interesting integral and sum representations, connection formulas, and error bounds.

The $q$-Laplace Transforms compared: the basic confluent hypergeometric function ${}_2φ_0$

TL;DR

The paper develops and compares three -Borel–Laplace resummations for divergent -Gevrey series arising from -difference equations, revealing that their differences are exactly -exponentially small and governed by universal Stokes-type structures. It then specializes to the simplest divergent basic hypergeometric function , introducing three resummed versions and deriving a comprehensive toolkit: integral, sum, Mellin–Barnes representations, and explicit connection formulas to the standard -Kummer solutions, together with sharp error bounds. A key result is that the -version, via Mellin–Barnes representations, is the most natural resummation, while the discrete -version offers a one-valued alternative with a family of poles. The work also yields a discrete orthogonality for Stieltjes–Wigert polynomials and highlights the interplay between three -Laplace transforms in driving -special function theory.

Abstract

In solving -difference equations, and in the definition of -special functions, we encounter formal power series in which the th coefficient is of size with fixed. To make sense of these formal series, a -Borel-Laplace resummation is required. There are three candidates for the -Laplace transform, resulting in three different resummations. Surprisingly, the differences between these resummations have not been discussed in the literature. Our main result provides explicit formulas for these -exponentially small differences. We also give simple Mellin--Barnes integral representations for all the basic hypergeometric functions and derive a third (discrete) orthogonality condition for the Stieltjes--Wigert polynomials. As the main application, we introduce three resummations for the functions which can be seen as versions of the Kummer functions. We derive many of their properties, including interesting integral and sum representations, connection formulas, and error bounds.

Paper Structure

This paper contains 10 sections, 9 theorems, 111 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and Summary
  2. $q$-Special Functions
  3. $q$-Laplace Transforms
  4. Application: the simplest divergent basic hypergeometric function
  5. Integral and Sum Representations
  6. Connection Relations
  7. Error Bounds
  8. Other Properties of ${\rm U}_q\left(a,b;z\right)$
  9. Cauchy-Heine Integral Representations
  10. Proofs

Key Result

Lemma 2.1

For $x,y,x/z,x/y\notin \left\{0,-1,-q^{\pm1},-q^{\pm2},\ldots \right\}$, we have

Figures (1)

  • Figure 1: Contour of integration $\mathcal{C}$ used to obtain the Cauchy-Heine integral representation.

Theorems & Definitions (17)

  • Lemma 2.1
  • proof
  • Theorem 3.1
  • Corollary 3.1
  • Theorem 5.1
  • Theorem 5.2
  • Theorem 5.3
  • Theorem 6.1
  • Corollary 6.1
  • Theorem 7.1
  • ...and 7 more