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The construction of $q$-analogues via $_3φ_2$-series and $q$-difference equations

John M. Campbell

TL;DR

The paper develops a multiparameter EKHAD-normalization framework built on the $q$-Zeilberger algorithm to construct $q$-WZ pairs for ${}_{3}\phi_{2}$-series, enabling telescoping to relate sums of the form $\sum_{k} F(0,k)$ and $\sum_{n} G(n,0)$. This approach generalizes EKHAD-normalization beyond fixed seeds, yielding broad families of $q$-analogues for accelerated hypergeometric series and Ramanujan-type evaluations of universal constants such as $\pi$ and $\Gamma$-values with rational arguments. By applying the method to Ramanujan-type, Fabry--Guillera, Apéry, Zeilberger, and BBP-type formulas, the work produces numerous new $q$-analogues and demonstrates a versatile route to generate further identities via parameter specialization. The results illuminate the unifying role of ${}_{3}\phi_{2}$-series in $q$-analogue accelerations and open pathways for combinatorial and number-theoretic applications of multiparameter $q$-WZ methods.

Abstract

We apply the EKHAD-normalization method given in our recent work to obtain, via the $q$-version of Zeilberger's algorithm, $q$-WZ pairs $(F, G)$ such that $\sum_{k = 0}^{\infty} F(0, k)$ may be expressed as a basic hypergeometric series of the form ${}_{3}φ_2$ with multiple free parameters, and in such a way so that $\sum_{k=0}^{\infty} F(0, k) = \sum_{n=0}^{\infty} G(n, 0)$. In contrast to how previous applications of EKHAD-normalization relied on $q$-analogues for specific WZ pairs introduced by Guillera, our multiparameter approach provides a broad framework in the construction of $q$-analogues for accelerated series for universal constants such as $π$. We apply this multiparameter version of EKHAD-normalization to obtain and prove new $q$-analogues for accelerated hypergeometric series attributed to many authors, including (alphabetically) Adamchik and Wagon, Apéry, Chu, Chu and Zhang, Fabry, Guillera, Ramanujan, and Zeilberger.

The construction of $q$-analogues via $_3φ_2$-series and $q$-difference equations

TL;DR

The paper develops a multiparameter EKHAD-normalization framework built on the -Zeilberger algorithm to construct -WZ pairs for -series, enabling telescoping to relate sums of the form and . This approach generalizes EKHAD-normalization beyond fixed seeds, yielding broad families of -analogues for accelerated hypergeometric series and Ramanujan-type evaluations of universal constants such as and -values with rational arguments. By applying the method to Ramanujan-type, Fabry--Guillera, Apéry, Zeilberger, and BBP-type formulas, the work produces numerous new -analogues and demonstrates a versatile route to generate further identities via parameter specialization. The results illuminate the unifying role of -series in -analogue accelerations and open pathways for combinatorial and number-theoretic applications of multiparameter -WZ methods.

Abstract

We apply the EKHAD-normalization method given in our recent work to obtain, via the -version of Zeilberger's algorithm, -WZ pairs such that may be expressed as a basic hypergeometric series of the form with multiple free parameters, and in such a way so that . In contrast to how previous applications of EKHAD-normalization relied on -analogues for specific WZ pairs introduced by Guillera, our multiparameter approach provides a broad framework in the construction of -analogues for accelerated series for universal constants such as . We apply this multiparameter version of EKHAD-normalization to obtain and prove new -analogues for accelerated hypergeometric series attributed to many authors, including (alphabetically) Adamchik and Wagon, Apéry, Chu, Chu and Zhang, Fabry, Guillera, Ramanujan, and Zeilberger.
Paper Structure (17 sections, 2 theorems, 129 equations)

This paper contains 17 sections, 2 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and background
  3. EKHAD-normalization
  4. Ramanujan-type formulas and $\Gamma$-values
  5. Motivating results
  6. Ramanujan's series of convergence rate $\frac{1}{4}$
  7. The Fabry--Guillera formula
  8. Apéry's series for $\zeta(2)$
  9. Zeilberger's series of convergence rate $\frac{1}{64}$
  10. A $q$-analogue of a BBP-type formula
  11. Multiparameter EKHAD-normalization
  12. $q$-analogues for the convergence rate $\frac{1}{4}$
  13. $q$-analogues for the convergence rate $-\frac{1}{4}$
  14. Further $q$-analogues for the convergence rate $\frac{1}{4}$
  15. $q$-analogues for the convergence rate $\frac{1}{64}$
  16. ...and 2 more sections

Key Result

Theorem 1

Letting $p(n) = q^{a+b+c+1}-q^{a+b+c+d+n} + q^{a+b+d+1} - q^{a+c+d+n+1}-q^{b+c+d+n+1}+q^{2 c+2 d+3 n}$, the relation holds if $| \frac{q^{a + b + 1}}{q^{c + d}} | < 1 < |q|$.

Theorems & Definitions (44)

  • Theorem 1
  • proof
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • ...and 34 more