$\,_{3}F_{4}$ hypergeometric functions as a sum of a product of $\,_{2}F_{3}$ functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911385375473664 |
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| author | Straton, Jack C. |
| author_facet | Straton, Jack C. |
| contents | This paper shows that certain $\,_{3}F_{4}$ hypergeometric functions can be expanded in sums of pair products of $\,_{1}F_{2}$ functions. In special cases, the $\,_{3}F_{4}$ hypergeometric functions reduce to $\,_{2}F_{3}$ functions. Further special cases allow one to reduce the $\,_{2}F_{3}$ functions to $\,_{1}F_{2}$ functions, and the sums to products of $\,_{0}F_{1}$ (Bessel) and $\,_{1}F_{2}$ functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions, $\,_{2}F_{1}$ functions, and $\,_{3}F_{2}$ functions into the realm of $\,_{p}F_{q}$ functions where $p<q$ for both the summand and terms in the series. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_06206 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\,_{3}F_{4}$ hypergeometric functions as a sum of a product of $\,_{2}F_{3}$ functions Straton, Jack C. General Mathematics 33C10, 42C10, 41A10, 33F10, 65D20, 68W30, 33D50, 33C05 This paper shows that certain $\,_{3}F_{4}$ hypergeometric functions can be expanded in sums of pair products of $\,_{1}F_{2}$ functions. In special cases, the $\,_{3}F_{4}$ hypergeometric functions reduce to $\,_{2}F_{3}$ functions. Further special cases allow one to reduce the $\,_{2}F_{3}$ functions to $\,_{1}F_{2}$ functions, and the sums to products of $\,_{0}F_{1}$ (Bessel) and $\,_{1}F_{2}$ functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions, $\,_{2}F_{1}$ functions, and $\,_{3}F_{2}$ functions into the realm of $\,_{p}F_{q}$ functions where $p<q$ for both the summand and terms in the series. |
| title | $\,_{3}F_{4}$ hypergeometric functions as a sum of a product of $\,_{2}F_{3}$ functions |
| topic | General Mathematics 33C10, 42C10, 41A10, 33F10, 65D20, 68W30, 33D50, 33C05 |
| url | https://arxiv.org/abs/2501.06206 |