Sum of a double series
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- by B. L. Sharma
- Proc. Amer. Math. Soc. 52 (1975), 136-138
- DOI: https://doi.org/10.1090/S0002-9939-1975-0387678-1
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Abstract:
In this paper we obtain the sum of a double series $F(1,1)$ and, in a particular case, we get a new formula $_4{F_3}(1)$, \[ _4{F_3}\left [ {\begin {array}{*{20}{c}} {\alpha ,\beta - \alpha ,1/2\rho ,1/2\rho + 1/2;1} \\ {1/2\beta ,1/2(1 + \beta ),1 + \rho ;} \\ \end {array} } \right ] = \tfrac {{\Gamma (\beta - \rho - \alpha )\Gamma (\beta )}} {{\Gamma (\beta - \rho )\Gamma (\beta - \alpha )}},\] provided that $R(\beta - \alpha ) > 0$, $R(\beta - \rho - \alpha ) > 0$ and $R(\beta - \rho ) > 0$. If $\alpha = - n$, the formula reduces to a known result due to Bailey [2].References
- P. Appell and J. Kampé de Feriet, Fonctions hypergéométriques et hypersphériques, polynômes d’Hermite, Gauthier-Villars, Paris, 1926.
W. N. Bailey, Some identities involving generalised hypergeometric series, Proc. London Math. Soc. (2) 29 (1929), 503-516.
- T. W. Chaundy, Expansions of hypergeometric functions, Quart. J. Math. Oxford Ser. 13 (1942), 159–171. MR 7819, DOI 10.1093/qmath/os-13.1.159
- L. Carlitz, A Saalschützian theorem for double series, J. London Math. Soc. 38 (1963), 415–418. MR 160944, DOI 10.1112/jlms/s1-38.1.415
- Leonard Carlitz, Summation of a double hypergeometric series, Matematiche (Catania) 22 (1967), 138–142. MR 214816
- R. N. Jain, Sum of a double hypergeometric series, Matematiche (Catania) 21 (1966), 300–301. MR 201700
- Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688
- B. L. Sharma, Summation of a double hypergeometric series, Matematiche (Catania) 28 (1973), 30–32 (1974). MR 352562
Bibliographic Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 52 (1975), 136-138
- MSC: Primary 33A30
- DOI: https://doi.org/10.1090/S0002-9939-1975-0387678-1
- MathSciNet review: 0387678