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Exponentially-improved asymptotics for $q$-difference equations: ${}_2φ_0$ and $q{\rm P}_{\rm I}$

Nalini Joshi, Adri Olde Daalhuis

Abstract

Usually when solving differential or difference equations via series solutions one encounters divergent series in which the coefficients grow like a factorial. Surprisingly, in the $q$-world the $n$th coefficient is often of the size $q^{-\frac12 n(n-1)}$, in which $q\in(0,1)$ is fixed. Hence, the divergence is much stronger, and one has to introduce alternative Borel and Laplace transforms to make sense of these formal series. We will discuss exponentially-improved asymptotics for the basic hypergeometric function ${}_2φ_0$ and for solutions of the $q$-difference first Painlevé equation $q{\rm P}_{\rm I}$. These are optimal truncated expansions, and re-expansions in terms of new $q$-hyperterminant functions. The re-expansions do incorporate the Stokes phenomena.

Exponentially-improved asymptotics for $q$-difference equations: ${}_2φ_0$ and $q{\rm P}_{\rm I}$

Abstract

Usually when solving differential or difference equations via series solutions one encounters divergent series in which the coefficients grow like a factorial. Surprisingly, in the -world the th coefficient is often of the size , in which is fixed. Hence, the divergence is much stronger, and one has to introduce alternative Borel and Laplace transforms to make sense of these formal series. We will discuss exponentially-improved asymptotics for the basic hypergeometric function and for solutions of the -difference first Painlevé equation . These are optimal truncated expansions, and re-expansions in terms of new -hyperterminant functions. The re-expansions do incorporate the Stokes phenomena.
Paper Structure (12 sections, 3 theorems, 89 equations, 1 figure)

This paper contains 12 sections, 3 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. $q$-special functions
  3. The $q$-Borel-Laplace transform
  4. The $q$-hyperterminant function
  5. Exponentially-improved asymptotics for ${}_2\phi_0$
  6. $q{\rm P}_{\rm I}$
  7. The Stokes multipliers
  8. The Stokes phenomenon and exponentially improved asymptotics
  9. Transseries analysis
  10. The order 1 solution and the distribution of poles and zeros
  11. Minor lemmas
  12. A simple nonlinear equation

Key Result

Lemma A.1

For the sequence $\{c_n\}$ defined by $c_0=1$ and recurrence relation crecur we have $\left|c_n\right|\leq \left(\frac{256}{27}\right)^n\left|q\right|^{-\frac{1}{2} n(n-1)}$ for $n=0,1,2,\ldots$ and $|q|\leq1$.

Figures (1)

  • Figure 1: The case $e_0=1$, $q=\frac{7}{10}$. The $\star$ are the locations of the double poles, the $\bullet$ are the simple zeros, and the $\times$ are stationary points, that is, solutions of $w(z)=z$.

Theorems & Definitions (6)

  • Lemma A.1
  • proof
  • Lemma A.2
  • proof
  • Lemma A.3
  • proof