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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2509.05897 |
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| _version_ | 1866916937911500800 |
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| author | Campbell, John M. |
| author_facet | Campbell, John M. |
| contents | The first known $q$-analogues for any of the $17$ formulas for $\frac{1}π$ due to Ramanujan were introduced in 2018 by Guo and Liu (J. Difference Equ. Appl. 29:505-513, 2018), via the $q$-Wilf-Zeilberger method. Through a "normalization" method, which we refer to as EKHAD-normalization, based on the $q$-polynomial coefficients involved in first-order difference equations obtained from the $q$-version of Zeilberger's algorithm, we introduce $q$-WZ pairs that extend WZ pairs introduced by Guillera (Adv. in Appl. Math. 29:599-603, 2002) (Ramanujan J. 11:41-48, 2006). We apply our EKHAD-normalization method to prove four new $q$-analogues for three of Ramanujan's formulas for $\frac{1}π$ along with $q$-analogues of Guillera's first two series for $\frac{1}{π^2}$. Our normalization method does not seem to have been previously considered in any equivalent way in relation to $q$-series, and this is substantiated through our survey on previously known $q$-analogues of Ramanujan-type series for $\frac{1}π$ and of Guillera's series for $\frac{1}{π^2}$. We conclude by showing how our method can be adapted to further extend Guillera's WZ pairs by introducing hypergeometric expansions for $\frac{1}{π^2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05897 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $q$-analogues of $π$-formulas due to Ramanujan and Guillera Campbell, John M. Classical Analysis and ODEs Combinatorics 33F10 The first known $q$-analogues for any of the $17$ formulas for $\frac{1}π$ due to Ramanujan were introduced in 2018 by Guo and Liu (J. Difference Equ. Appl. 29:505-513, 2018), via the $q$-Wilf-Zeilberger method. Through a "normalization" method, which we refer to as EKHAD-normalization, based on the $q$-polynomial coefficients involved in first-order difference equations obtained from the $q$-version of Zeilberger's algorithm, we introduce $q$-WZ pairs that extend WZ pairs introduced by Guillera (Adv. in Appl. Math. 29:599-603, 2002) (Ramanujan J. 11:41-48, 2006). We apply our EKHAD-normalization method to prove four new $q$-analogues for three of Ramanujan's formulas for $\frac{1}π$ along with $q$-analogues of Guillera's first two series for $\frac{1}{π^2}$. Our normalization method does not seem to have been previously considered in any equivalent way in relation to $q$-series, and this is substantiated through our survey on previously known $q$-analogues of Ramanujan-type series for $\frac{1}π$ and of Guillera's series for $\frac{1}{π^2}$. We conclude by showing how our method can be adapted to further extend Guillera's WZ pairs by introducing hypergeometric expansions for $\frac{1}{π^2}$. |
| title | $q$-analogues of $π$-formulas due to Ramanujan and Guillera |
| topic | Classical Analysis and ODEs Combinatorics 33F10 |
| url | https://arxiv.org/abs/2509.05897 |