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Basic hypergeometric identities derived from three-term relations

Yuka Yamaguchi

TL;DR

This work develops a $q$-analogue of Ebisu's three-term method to derive basic hypergeometric identities from linear relations among ${}_{2}\phi_{1}$ series. By analyzing $Q^{(N)}$ and $R^{(N)}$ in carefully chosen index quadruples $(k,l,m,n)$, the authors obtain first-order $q$-difference equations whose solutions yield identities such as the $q$-binomial theorem, Heine's $q$-Gauss summation, and $q$-Kummer-type results, along with Ebisu-type identities sv1--sv4 and a generalized sv5. The paper then studies the symmetries of the coefficients, establishing a 96-element symmetry group $G$ acting on $(k,l,m,n;a,b,c,x)$ and showing a complete set of representatives for the quotient $\\mathbb{Z}^4 / G'$, enabling systematic generation of identities. Together, these results provide a structured framework for deriving and organizing a broad class of basic hypergeometric identities, with conjectures for further generalizations in section 5. The approach combines $q$-shifted factorials, Heine transformations, and group-theoretic symmetries to extend Ebisu’s method into the basic hypergeometric realm.

Abstract

In 2015, Ebisu presented a new method for finding hypergeometric identities based on three-term relations for the ${}_{2} F_{1}$ hypergeometric series. By using this method, he derived almost all of the previously known hypergeometric identities, as well as many new ones. In this paper, we derive several basic hypergeometric identities, including both well-known and not widely known ones, by applying a $q$-analogue of Ebisu's method to three-term relations for the ${}_{2} φ_{1}$ basic hypergeometric series.

Basic hypergeometric identities derived from three-term relations

TL;DR

This work develops a -analogue of Ebisu's three-term method to derive basic hypergeometric identities from linear relations among series. By analyzing and in carefully chosen index quadruples , the authors obtain first-order -difference equations whose solutions yield identities such as the -binomial theorem, Heine's -Gauss summation, and -Kummer-type results, along with Ebisu-type identities sv1--sv4 and a generalized sv5. The paper then studies the symmetries of the coefficients, establishing a 96-element symmetry group acting on and showing a complete set of representatives for the quotient , enabling systematic generation of identities. Together, these results provide a structured framework for deriving and organizing a broad class of basic hypergeometric identities, with conjectures for further generalizations in section 5. The approach combines -shifted factorials, Heine transformations, and group-theoretic symmetries to extend Ebisu’s method into the basic hypergeometric realm.

Abstract

In 2015, Ebisu presented a new method for finding hypergeometric identities based on three-term relations for the hypergeometric series. By using this method, he derived almost all of the previously known hypergeometric identities, as well as many new ones. In this paper, we derive several basic hypergeometric identities, including both well-known and not widely known ones, by applying a -analogue of Ebisu's method to three-term relations for the basic hypergeometric series.

Paper Structure

This paper contains 10 sections, 3 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Derivation of identities
  3. Proof of Theorem $\mathbf{\ref{['thm:1']}}$
  4. Generalization of the last identity in Theorem $\mathbf{\ref{['thm:1']}}$
  5. Symmetries of the coefficients
  6. Proof of Lemma $\mathbf{\ref{['lem:G']}}$
  7. Proof of Theorem $\mathbf{\ref{['thm:3']}}$
  8. Symmetries of another coefficient: additional remarks
  9. Complete set of representatives for $\mathbf{\mathbb{Z}^4 / G'}$
  10. Considerations and conjectures for more general cases

Key Result

Theorem 1

When $(k,l,m,n)=(0,0,0,2)$, the three-term relation $(3tr)$ leads to the identity When $(k,l,m,n)=(0,1,1,0)$, the relation $(3tr)$ yields When $(k,l,m,n)=(0,2,2,0)$, the relation $(3tr)$ gives When $(k,l,m,n)=(1,2,1,-1)$, the relation $(3tr)$ leads to the identities When $(k,l,m,n)=(0,3,3,0)$, the relation $(3tr)$ yields In the above identities, $M$ and $N$ denote arbitrary non-negative integ

Theorems & Definitions (5)

  • Theorem 1
  • Lemma 2
  • Theorem 3
  • Remark 4
  • Remark 5