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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2405.02776 |
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| _version_ | 1866916235474632704 |
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| author | Campbell, John M. |
| author_facet | Campbell, John M. |
| contents | Chu and Zhang, in 2014, introduced hypergeometric transforms derived through the application of an Abel-type summation lemma to Dougall's ${}_{5}H_{5}$-series. These transforms were applied by Chu and Zhang to obtain accelerated rates of convergence, yielding rational series related to the work of Ramanujan and Guillera. We apply a variant of an acceleration method due to Wilf using what we refer to as shifted indices for Pochhammer symbols involved in our first-order, inhomogeneous recurrences derived via Zeilberger's algorithm, to build upon Chu and Zhang's accelerations, recovering many of their accelerated series and introducing many inequivalent series for universal constants, including series of Ramanujan type involving linear polynomials as summand factors, as in Ramanujan's series for $\frac{1}π$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02776 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hypergeometric accelerations with shifted indices Campbell, John M. Classical Analysis and ODEs 33F10 Chu and Zhang, in 2014, introduced hypergeometric transforms derived through the application of an Abel-type summation lemma to Dougall's ${}_{5}H_{5}$-series. These transforms were applied by Chu and Zhang to obtain accelerated rates of convergence, yielding rational series related to the work of Ramanujan and Guillera. We apply a variant of an acceleration method due to Wilf using what we refer to as shifted indices for Pochhammer symbols involved in our first-order, inhomogeneous recurrences derived via Zeilberger's algorithm, to build upon Chu and Zhang's accelerations, recovering many of their accelerated series and introducing many inequivalent series for universal constants, including series of Ramanujan type involving linear polynomials as summand factors, as in Ramanujan's series for $\frac{1}π$. |
| title | Hypergeometric accelerations with shifted indices |
| topic | Classical Analysis and ODEs 33F10 |
| url | https://arxiv.org/abs/2405.02776 |