Y S KİM, A K RATHİE, R. B. PARİS, U PANDEY

SOME SUMMATION FORMULAS FOR THE HYPERGEOMETRIC SERIES

The aim of this paper is to obtain explicit expressions of the generalizedhypergeometric function r+2Fr+1a, b, 1 2 (a + b + j + 1), (fr + mr) (fr) ; 1 2 for j = 0, ±1, . . . , ±5, where r pairs of numeratorial and denominatorialparameters differ by positive integers mr. The results are derived withthe help of an expansion in terms of a finite sum of 2F1( 1 2 ) functions anda generalization of Gauss' second summation theorem due to Lavoie etal. [J. Comput. Appl. Math. 72, 293-300 (1996)]. Some special andlimiting cases are also given.

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[1] Abramowitz, M. and Stegun, I. A. (Eds.) Handbook of Mathematical Functions (Dover, New York, 1965).
[2] Karlsson, P. W. Hypergeometric functions with integral parameter differences, J. Math. Phys. 12, 270-271, 1971.
[3] Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Watson's theorem on the sum of a 3F2, Indian J. Math. 34, 23-32, 1992.
[4] Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Dixon's theorem on the sum of a 3F2, Math. Comp. 62, 267-276, 1994.
[5] Lavoie, J. L., Grondin, F. and Rathie, A. K. Generalizations of Whipple's theorem on the sum of a 3F2, J. Comput. Appl. Math. 72, 293-300, 1996.
[6] Luke, Y. L. Mathematical Functions and Their Approximations (Academic Press, New York, 1975).
[7] Miller, A. R. Certain summation and transformation formulas for generalized hypergeometric series, J. Comput. Appl. Math. 231, 964-972, 2009.
[8] Miller, A. R. and Paris, R. B. Certain transformations and summations for the generalized hypergeometric series with integral parameter differences, Integral Transforms and Special Functions, 22, 67-77, 2011.
[9] Miller, A. R. and Paris, R. B. Euler-type transformations for the generalized hypergeometric function r+2Fr+1(x), Zeitschrift angew. Math. Phys., 62, 31-45, 2011.
[10] Miller, A. R. and Paris, R. B. On a result related to transformations and summations of generalized hypergeometric series, Math. Communications, 17, 205-210, 2012.
[11] Miller, A. R. and Paris, R. B. Transformation formulas for the generalized hypergeometric function with integral parameter differences, Rocky Mountain J. Math., 43, 291-327, 2013.
[12] Miller, A. R. and Srivastava, H. M. Karlsson-Minton summation theorems for the generalized hypergeometric series of unit argument, Integral Transforms and Special Functions, 21, 603-612, 2010.
[13] Minton, B. M. Generalized hypergeometric function of unit argument, J. Math. Phys., 11, 1375-1376, 1970.
[14] Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I. Integrals and Series: More Special Functions, vol. 3 (Gordon and Breach, New York, 1990).
[15] Rathie, A. K. and Pog´any, T. K. New summation formula for 3F2( 1 2 ) and a Kummer-type II transformation of 2F2(x), Mathematical Communications, 13, 63-66, 2008.
[16] Slater, L. J. Generalized Hypergeometric Functions (Cambridge University Press, Cambridge, 1966).