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Degenerations of $q$-Heun equation

Chihiro Sato, Kouichi Takemura

TL;DR

The work analyzes degenerations of the $q$-Heun equation by exploiting linear $q$-difference Lax pairs associated to various $q$-Painlevé equations. By applying Murata’s and Kajiwara–Noumi–Yamada’s Lax structures and suitable gauge transformations, it derives explicit degenerated $q$-Heun equations, identifying confluent, biconfluent, and doubly confluent families and their reduced variants. The authors also establish limit procedures to the corresponding differential equations as $q\to1$, connecting the $q$-deformations to classical confluent Heun equations. The results provide a systematic bridge between $q$-Painlevé degenerations and Heun-type equations, with implications for both exact solvability and asymptotic analysis in $q$-difference and differential settings.

Abstract

We obtain several degenerations of the $q$-Heun equation by considering the linear $q$-difference equations associated to several $q$-Painlevé equations. We establish definitions of the confluent $q$-Heun equation, the biconfluent $q$-Heun equation and the doubly confluent $q$-Heun equation, and investigate limit procedures to the corresponding differential equations.

Degenerations of $q$-Heun equation

TL;DR

The work analyzes degenerations of the -Heun equation by exploiting linear -difference Lax pairs associated to various -Painlevé equations. By applying Murata’s and Kajiwara–Noumi–Yamada’s Lax structures and suitable gauge transformations, it derives explicit degenerated -Heun equations, identifying confluent, biconfluent, and doubly confluent families and their reduced variants. The authors also establish limit procedures to the corresponding differential equations as , connecting the -deformations to classical confluent Heun equations. The results provide a systematic bridge between -Painlevé degenerations and Heun-type equations, with implications for both exact solvability and asymptotic analysis in -difference and differential settings.

Abstract

We obtain several degenerations of the -Heun equation by considering the linear -difference equations associated to several -Painlevé equations. We establish definitions of the confluent -Heun equation, the biconfluent -Heun equation and the doubly confluent -Heun equation, and investigate limit procedures to the corresponding differential equations.
Paper Structure (20 sections, 1 theorem, 90 equations)

This paper contains 20 sections, 1 theorem, 90 equations.

Table of Contents

  1. Introduction
  2. Newton polygon and gauge transformation
  3. Degenerations of $q$-Heun equations arising from the list of Murata
  4. $q$-P($A_4$)
  5. $q$-P($A_5$)
  6. $q$-P$(A_5)^\#$
  7. $q$-P($A_6$)
  8. $q$-P$(A_6)^\#$
  9. $q$-P($A_7$)
  10. $q$-P($A_7'$)
  11. Degenerations of $q$-Heun equations arising from the list of Kajiwara-Noumi-Yamada
  12. $q$-P($A_4^{(1)}$)
  13. $q$-P($E_3^{(1)};a$)
  14. $q$-P($E_3^{(1)};b$)
  15. $q$-P($E_2^{(1)};a$)
  16. ...and 5 more sections

Key Result

Proposition 2.1

$$ (i) If $y(x)$ satisfies $q^{-\lambda}a(x)g(x/q)+b(x)g(x)+q^{\lambda}c(x)g(qx)=0$, then the function $h(x)=x^{\lambda}y(x)$ satisfies $a(x)g(x/q)+b(x)g(x)+c(x)g(qx)=0$. (ii) If $y(x)$ satisfies $(1-\alpha x)a(x)g(x/q)+b(x)g(x)+c(x)g(qx)=0$, then the function $u(x)=(q\alpha x;q)_{\infty}y(x)$ satis

Theorems & Definitions (1)

  • Proposition 2.1