Abstract
By combining the series rearrangement with a reformulation of Beta function, we prove a transformation theorem that reduces a double sum with seven free parameters to a terminating hypergeometric \({_4F_3}\)-series. Seven double sum identities are derived as consequences with one of them extending the recent double sum of Dunkl and Gasper (The sums of a double hypergeometric series and of the first m + 1 terms of \({_3F_2(a,b,c;(a+b+1)/2,2c;1)}\) when c = −m is a negative integer. arXiv:1412.4022v2 [math.CA], 2014).
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References
Bailey W.N.: Transformations of generalized hypergeometric series. Proc. Lond. Math. Soc. 29(2), 495–502 (1929)
Bailey W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935)
Bailey W.N.: On the sum of a terminating \({_3F_2(1)}\). Q. J. Math. Oxf. (Ser. 2) 4, 237–240 (1953)
Chu W.: Inversion techniques and combinatorial identities: strange evaluations of hypergeometric series. Pure Math. Appl. 4(4), 409–428 (1993)
Chu W.: Inversion techniques and combinatorial identities: a quick introduction to hypergeometric evaluations. Math. Appl. 283, 31–57 (1994)
Chu W., Srivastava H.M.: Ordinary and basic bivariate hypergeometric transformations associated with the Appell and Kampé de Fériet functions. J. Comput. Appl. Math. 156, 355–370 (2003)
Chu W., Wang X.: Whipple-like formulae for terminating \({_3F_2}\)-series. Util. Math. 93, 65–78 (2014)
Chu, W., Zhou, R.R.: Watson–like formulae for terminating \({_3F_2}\)-series. In: Kotsireas, I.S., Zima, E.V. (eds.) Advances in Combinatorics (in Memory of Herbert S. Wilf). Springer, Berlin, Heidelberg, pp. 139–159 (2013)
Dixon A.C.: Summation of a certain series. Proc. Lond. Math. Soc. 35(1), 285–289 (1903)
Dunkl, C.F., Gasper, G.: The sums of a double hypergeometric series and of the first m + 1 terms of \({_3F_2(a,b,c;(a+b+1)/2,2c;1)}\) when c = −m is a negative integer (2014). arXiv:1412.4022v2 [math.CA]
Dunkl C.F., Slater P.B.: Separability probability formulas and their proofs for generalized two–qubit X-matrices endowed with Hilbert–Schmidt and induced measures. Random Matrices Theory Appl. 4(4), 1550018 (2015)
Gould H.W., Hsu L.C.: Some new inverse series relations. Duke Math. J. 40, 885–891 (1973)
NIST: NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/16.4#iii
Watson G.N.: A note on generalized hypergeometric series. Proc. London Math. Soc. 23(2), xiii–xv (1925)
Whipple F.J.W.: A group of generalized hypergeometric series: relations between 120 allied series of the type F[a,b,c;d,e]. Proc. Lond. Math. Soc. 23(2), 104–114 (1925)
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Chu, W. Further Double Sums of Dunkl and Gasper. Results Math 72, 171–180 (2017). https://doi.org/10.1007/s00025-016-0594-z
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DOI: https://doi.org/10.1007/s00025-016-0594-z