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Solutions to variant of $q$-hypergeometric equation of degree 2 associated with $q$-middle convolution

Yumi Arai

Abstract

We investigate the integral representations of solutions to the variant of $q$-hypergeometric equation of degree 2 obtained through $q$-middle convolution by using transformation formulas for $q$-hypergeometric series. We show the correspondence between these integral solutions and solutions obtained by other methods. We also show the linear relationships among the integral solutions.

Solutions to variant of $q$-hypergeometric equation of degree 2 associated with $q$-middle convolution

Abstract

We investigate the integral representations of solutions to the variant of -hypergeometric equation of degree 2 obtained through -middle convolution by using transformation formulas for -hypergeometric series. We show the correspondence between these integral solutions and solutions obtained by other methods. We also show the linear relationships among the integral solutions.
Paper Structure (9 sections, 71 equations)

This paper contains 9 sections, 71 equations.

Table of Contents

  1. Introduction
  2. Integral representation of solutions to variant of $q$-hypergeometric equation of degree 2
  3. Correspondence with already found solutions
  4. Correspondence with $q$-Appell series solution found in HMSTMST
  5. Correspondence with solutions obtained by Fujii and Nobukawa
  6. Linear relationships among integral solutions associated with $q$-middle convolution
  7. Concluding remarks
  8. Supplement to Remark \ref{['rem:b1b2xm=0']}
  9. Supplement to Theorem \ref{['thm:rlm=0']}

Theorems & Definitions (3)

  • proof
  • proof
  • proof