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GENERALIZATIONS OF TWO SUMMATION FORMULAS FOR THE GENERALIZED HYPERGEOMETRIC FUNCTION OF HIGHER ORDER DUE TO EXTON

Choi, June-Sang;Rathie, Arjun Kumar

  • Received : 2009.03.11
  • Published : 2010.07.31

Abstract

In 1997, Exton, by mainly employing a widely-used process of resolving hypergeometric series into odd and even parts, obtained some new and interesting summation formulas with arguments 1 and -1. We aim at showing how easily many summation formulas can be obtained by simply combining some known summation formulas. Indeed, we present 22 results in the form of two generalized summation formulas for the generalized hypergeometric series $_4F_3$, including two Exton's summation formulas for $_4F_3$ as special cases.

Keywords

generalized hypergeometric series $_pF_q$;summation theorems for $_pF_q$

References

  1. J. Choi and H. M. Srivastava, Certain classes of infinite series, Monatsh. Math. 127 (1999), no. 1, 15–25. https://doi.org/10.1007/s006050050019
  2. W. Chu and L. de Donno, Hypergeometric series and harmonic number identities, Adv. in Appl. Math. 34 (2005), no. 1, 123–137. https://doi.org/10.1016/j.aam.2004.05.003
  3. H. Exton, Multiple Hypergeometric Functions and Applications, Ellis Horwood, Chichester, UK, 1976.
  4. H. Exton, Some new summation formulae for the generalised hypergeometric function of higher order, J. Comput. Appl. Math. 79 (1997), no. 2, 183–187. https://doi.org/10.1016/S0377-0427(96)00154-9
  5. J. L. Lavoie, F. Grondin, and A. K. Rathie, Generalizations of Whipple’s theorem on the sum of a $_3F_2$, J. Comput. Appl. Math. 72 (1996), no. 2, 293–300. https://doi.org/10.1016/0377-0427(95)00279-0
  6. E. D. Rainville, Special Functions, The Macmillan Company, New York, 1960.
  7. L.-C. Shen, Remarks on some integrals and series involving the Stirling numbers and ${\zeta}(n)$, Trans. Amer. Math. Soc. 347 (1995), no. 4, 1391–1399. https://doi.org/10.2307/2154819
  8. L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.
  9. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston, and London, 2001.