Sum of a double series
Author:
B. L. Sharma
Journal:
Proc. Amer. Math. Soc. 52 (1975), 136138
MSC:
Primary 33A30
DOI:
https://doi.org/10.1090/S00029939197503876781
MathSciNet review:
0387678
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Abstract  References  Similar Articles  Additional Information
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].

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 T. W. Chaundy, Expansions of hypergeometric functions, Quart. J. Math. Oxford Ser. 13 (1942), 159–171. MR 7819, DOI https://doi.org/10.1093/qmath/os13.1.159
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 Leonard Carlitz, Summation of a double hypergeometric series, Matematiche (Catania) 22 (1967), 138–142. MR 214816
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Additional Information
Keywords:
Double series,
Saalschutizian theorem,
hypergeometric series of higher order and of two variables,
summation formula
Article copyright:
© Copyright 1975
American Mathematical Society