Mohammad Idris QURESHI, Shakir Hussain MALIK, Richard Bruce PARIS

A contiguous extension of Dixon’s theorem for a terminating 4F3(1) series with applications

We derive a summation formula for the terminating hypergeometric series 4F3 [ −m, a, b, 1 + c 1 + a + m, 1 + a − b, c ; 1] , where m denotes a nonnegative integer. Using this summation formula, we establish a reduction formula for the Srivastava–Daoust double hypergeometric function with arguments z and −z . Special cases of this reduction formula lead to several reduction formulas for the hypergeometric functions p+1Fp with quadratic arguments when p = 2, 3 and 4 by employing series rearrangement techniques. A general double series identity involving a bounded sequence of arbitrary complex numbers is also given.

PDF

___

[1] Andrews GE, Askey R, Roy R. Special Functions. Cambridge, UK: Cambridge University Press, 1999.
[2] Appell P, Kampé de Fériet J. Fonctions Hypergéométriques et Hypersphériques: Polynômes d’Hermite. Paris, France: Gauthier-Villars, 1926 (in French).
[3] Kim YS, Rakha MA, Rathie AK. Extensions of certain classical summation theorems for the series 2F1, 3F2 and 4F3 with applications in Ramanujan’s summations. International Journal of Mathematics and Mathematical Sciences 2010; 2010: 1-26. doi:10.1155/2010/309503
[4] Miller AR. A summation formula for Clausen’s series 3F2(1) with an application to Goursat’s function 2F2(x). Journal of Physics A: Mathematical and General 2005; 38: 3541-3545. doi: 10.1088/0305-4470/38/16/005
[5] Miller AR. Certain summation and transformation formulas for generalized hypergeometric series. Journal of Computational and Applied Mathematics 2009; 231: 964-972. doi: 10.1016/j.cam.2009.05.013
[6] Miller AR, Paris RB. Certain transformations and summations for generalized hypergeometric series with integral parameter differences. Integral Transforms and Special Functions 2011; 22 (1): 67-77. doi: 10.1080/10652469.2010.498001
[7] Miller AR, Paris RB. Euler-type transformations for the generalized hypergeometric function r+2Fr+1(x). Zeitschrift für Angewandte Mathematik und Physik 2011; 62: 31-45. doi: 10.1007/s00033-010-0085-0
[8] Miller AR, Paris RB. Transformation formulas for the generalized hypergeometric function with integral parameter differences. Rocky Mountain Journal of Mathematics 2013; 43 (1): 291-327. doi:10.1216/RMJ-2013-43-1-291
[9] Miller AR, Srivastava HM. Karlsson-Minton summation theorems for the generalized hypergeometric series of unit argument. Integral Transforms and Special Functions 2010; 21 (8): 603-612. doi: 10.1080/10652460903497259
[10] Prudnikov AP, Brychkov YuA, Marichev OI. Integrals and Series, Volume 3: More special functions. Moscow, Rusia: Nauka, 1986; Translated from the Russian by G. G. Gould, New York, NY, USA: Gordon and Breach Science Publishers, 1990.
[11] Rainville ED. Special Functions. New York, NY, USA: The Macmillan Company, 1960; Reprinted by Bronx, New York, NY, USA: Chelsea Publishing Company, 1971.
[12] Slater LJ. Generalized Hypergeometric Functions. Cambridge, UK: Cambridge University Press, 1966.
[13] Srivastava HM, Daoust MC. On Eulerian integrals associated with Kampé de Fériet’s function. Publications De L’Institut Mathématique Nouvelle Série, tome 1969; 9 (23): 199-202.
[14] Srivastava HM, Daoust MC. Certain generalized Neumann expansions associated with the Kampé de Fériet’s function. Koninklijke Nederlandse Akademie van Wetenschappen Proceedings Series A, 72-Indagationes Mathematicae 1969; 31: 449-457.
[15] Srivastava HM, Daoust MC. A note on the convergence of Kampé de Fériet’s double hypergeometric series. Mathematische Nachrichten 1972; 53: 151-159. doi: 10.1002/mana.19720530114 [16] Srivastava HM, Manocha HL. A Treatise on Generating Functions. Chichester, UK: Ellis Horwood Limited Publisher, 1984.