Summed series involving \,_{1}F_{2} hypergeometric functions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912832356876288 |
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| author | Straton, Jack C. |
| author_facet | Straton, Jack C. |
| contents | In a prior paper we found that the Fourier-Legendre series of a Bessel function of the first kind J_{N}\left(kx\right) and of a modified Bessel functions of the first kind I_{N}\left(kx\right) lead to an infinite set of series involving \,_{1}F_{2} hypergeometric functions (extracted therefrom) that could be summed, having values that are inverse powers of the eight primes 1/\left(2^{i}3^{j}5^{k}7^{l}11^{m}13^{n}17^{o}19^{p}\right) multiplying powers of the coefficient k, for the first 22 terms in each series. The present paper shows how to generate additional, doubly infinite summed series involving \,_{1}F_{2} hypergeometric functions from Chebyshev polynomial expansions of Bessel functions, and trebly infinite sets of summed series involving \,_{1}F_{2} hypergeometric functions from Gegenbauer polynomial expansions of Bessel functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00019 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Summed series involving \,_{1}F_{2} hypergeometric functions Straton, Jack C. General Mathematics 33C10, 42C10, 41A10, 41A50, 33F10, 65D20, 68W30 In a prior paper we found that the Fourier-Legendre series of a Bessel function of the first kind J_{N}\left(kx\right) and of a modified Bessel functions of the first kind I_{N}\left(kx\right) lead to an infinite set of series involving \,_{1}F_{2} hypergeometric functions (extracted therefrom) that could be summed, having values that are inverse powers of the eight primes 1/\left(2^{i}3^{j}5^{k}7^{l}11^{m}13^{n}17^{o}19^{p}\right) multiplying powers of the coefficient k, for the first 22 terms in each series. The present paper shows how to generate additional, doubly infinite summed series involving \,_{1}F_{2} hypergeometric functions from Chebyshev polynomial expansions of Bessel functions, and trebly infinite sets of summed series involving \,_{1}F_{2} hypergeometric functions from Gegenbauer polynomial expansions of Bessel functions. |
| title | Summed series involving \,_{1}F_{2} hypergeometric functions |
| topic | General Mathematics 33C10, 42C10, 41A10, 41A50, 33F10, 65D20, 68W30 |
| url | https://arxiv.org/abs/2412.00019 |