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| Formato: | Preprint |
| Publicado: |
2023
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2301.09699 |
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| _version_ | 1866915194937016320 |
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| author | Ulgenes, David Peter Hadrian |
| author_facet | Ulgenes, David Peter Hadrian |
| contents | We derive product and series representations of the gamma function using Newton interpolation series. Using these identities, a new formula for the coefficients in the Taylor series of the reciprocal gamma function is found. We also find two new series representations for the Euler-Mascheroni constant, containing only rational terms. After that, we introduce a new pseudogamma function which we call the $Λ$ function. This function interpolates the factorial at the positive integers, the reciprocal factorial at the negative integers, and is convergent for the entire real axis. Finally, we conjecture a novel series representation for the principal branch of the inverse gamma function $\text{inv}Γ_0(z)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_09699 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Series and Product Representations of Gamma, Pseudogamma and Inverse Gamma Functions Ulgenes, David Peter Hadrian Number Theory We derive product and series representations of the gamma function using Newton interpolation series. Using these identities, a new formula for the coefficients in the Taylor series of the reciprocal gamma function is found. We also find two new series representations for the Euler-Mascheroni constant, containing only rational terms. After that, we introduce a new pseudogamma function which we call the $Λ$ function. This function interpolates the factorial at the positive integers, the reciprocal factorial at the negative integers, and is convergent for the entire real axis. Finally, we conjecture a novel series representation for the principal branch of the inverse gamma function $\text{inv}Γ_0(z)$. |
| title | Series and Product Representations of Gamma, Pseudogamma and Inverse Gamma Functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2301.09699 |