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Jackson integral representation for a multiple $q$-hypergeometric series and an extension of the $q$-Riemann-Papperitz system

Takahiko Nobukawa

TL;DR

This paper addresses representing Kajihara's multivariable $q$-hypergeometric series $W^{M,2}$ via a Jackson integral and constructing a corresponding multivariable $q$-difference system $q$-$RP^M$. The authors derive a Jackson integral representation for $W^{M,2}$ and formulate a rank $(M+1)$ $q$-difference system that extends HMST's variant toward a $q$-Riemann–Papperitz framework, with Bailey-type transformations recovering known one-variable results when $M=1$. They further show that $q$-$RP^M$ degenerates to the $q$-Appell–Lauricella system, obtaining degenerate solutions in terms of the Appell–Lauricella function $oldsymbol{ extphi}_D$ and establishing transformation relations between these solutions. The results advance the theory of multivariable $q$-hypergeometric integrals and their associated differential/difference systems, with potential applications to integrable systems, Koornwinder operators, and related $q$-Garnier-type dynamics.

Abstract

We give a Jackson integral representation for Kajihara's $q$-hypergeometric series $W^{M,2}$. We construct a $q$-difference system that corresponds to this integral. This system is an extension of the variant of $q$-hypergeometric equation of degree three, defined by Hatano-Matsunawa-Sato-Takemura. We show that this system includes the $q$-Appell-Lauricella system as a degeneration.

Jackson integral representation for a multiple $q$-hypergeometric series and an extension of the $q$-Riemann-Papperitz system

TL;DR

This paper addresses representing Kajihara's multivariable -hypergeometric series via a Jackson integral and constructing a corresponding multivariable -difference system -. The authors derive a Jackson integral representation for and formulate a rank -difference system that extends HMST's variant toward a -Riemann–Papperitz framework, with Bailey-type transformations recovering known one-variable results when . They further show that - degenerates to the -Appell–Lauricella system, obtaining degenerate solutions in terms of the Appell–Lauricella function and establishing transformation relations between these solutions. The results advance the theory of multivariable -hypergeometric integrals and their associated differential/difference systems, with potential applications to integrable systems, Koornwinder operators, and related -Garnier-type dynamics.

Abstract

We give a Jackson integral representation for Kajihara's -hypergeometric series . We construct a -difference system that corresponds to this integral. This system is an extension of the variant of -hypergeometric equation of degree three, defined by Hatano-Matsunawa-Sato-Takemura. We show that this system includes the -Appell-Lauricella system as a degeneration.
Paper Structure (16 sections, 22 theorems, 207 equations)

This paper contains 16 sections, 22 theorems, 207 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Integral representation for $W^{M,2}$ and corresponding $q$-difference system
  4. Integral representation for $W^{M,2}$
  5. A $q$-difference system
  6. Degeneration to the $q$-Appell--Lauricella system
  7. Summary and discussions
  8. Appendix
  9. Convergence of Kajihara's series \ref{['defkaji']}
  10. Justification of the limit \ref{['just1']}: $k\to\infty$
  11. Justification of the limits \ref{['just2']} and \ref{['just3']}: $k\to\infty$
  12. Justification of the limit \ref{['just4']}: $a_{M+3}\to\infty$
  13. Justification of the limit \ref{['just5']}: $a_{M+2}\to0$
  14. Justification of the limits \ref{['just6']} and \ref{['just7']}: $a_{M+3}\to\infty$
  15. Justification of the limit \ref{['just8']}: $a_{M+2}\to0$
  16. ...and 1 more sections

Key Result

Proposition 2.5

For any $n\in\mathbb{Z}_{\geq 0}$, we have where $\mu=a^{N+2} q^{N+1}y_1 y_2 \cdots y_N/(c^{N+1} b_1 b_2 \cdots b_{M+N+2} x_1 x_2 \cdots x_M )$.

Theorems & Definitions (62)

  • Definition 2.1: GR
  • Definition 2.2: Kajihara's $q$-hypergeometric series Kajikaji2018KN
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5: Kajihara's transformation formula Kajikaji2018KN
  • Corollary 2.6: Kaji
  • proof
  • Remark 2.7
  • Theorem 3.1
  • Remark 3.2
  • ...and 52 more