The construction of $q$-analogues via $_3ϕ_2$-series and $q$-difference equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915799743070208 |
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| author | Campbell, John M. |
| author_facet | Campbell, John M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14314 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The construction of $q$-analogues via $_3ϕ_2$-series and $q$-difference equations Campbell, John M. Combinatorics 33F10, 39A13, 33D15, 33D05 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. |
| title | The construction of $q$-analogues via $_3ϕ_2$-series and $q$-difference equations |
| topic | Combinatorics 33F10, 39A13, 33D15, 33D05 |
| url | https://arxiv.org/abs/2602.14314 |