Evaluation of certain families of log-cosine integrals using hypergeometric function approach and applications
In this paper, we provide the analytical solutions of the families of certain definite integrals: $\int_0^\pi x^{m}\{\ln(2\cos\frac{x}{2})\}^{n}dx$ $(m\in\mathbb{N}_{0}$ and $n\in\mathbb{N}),$ in terms of multiple hypergeometric functions of Kampé de Fériet having the arguments $\pm1$ and Riemann ze...
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| Published in: | Notes on number theory and discrete mathematics Vol. 30; no. 3; pp. 499 - 515 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
"Prof. Marin Drinov" Publishing House of Bulgarian Academy of Sciences
01.10.2024
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| Subjects: | |
| ISSN: | 1310-5132, 2367-8275 |
| Online Access: | Get full text |
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| Summary: | In this paper, we provide the analytical solutions of the families of certain definite integrals: $\int_0^\pi x^{m}\{\ln(2\cos\frac{x}{2})\}^{n}dx$ $(m\in\mathbb{N}_{0}$ and $n\in\mathbb{N}),$ in terms of multiple hypergeometric functions of Kampé de Fériet having the arguments $\pm1$ and Riemann zeta functions. As applications, we obtain some mixed summation formulas (19), (35) and (46) involving generalized hypergeometric functions $_3F_2,$ $_5F_4$ and $_7F_6$ having the arguments $\pm 1$ and other (possibly) new summation formulas (38) and (40) for multiple hypergeometric functions of Kampé de Fériet having the arguments $\pm 1$ also mixed relations (36) and (47) involving Riemann zeta functions. |
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| ISSN: | 1310-5132 2367-8275 |
| DOI: | 10.7546/nntdm.2024.30.3.499-515 |