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ON A NEW CLASS OF SERIES IDENTITIES

  • SHEKHAWAT, NIDHI (Research Scholar, Department of Mathematics & Statistics, Bansthali University) ;
  • CHOI, JUNESANG (Department of Mathematics, Dongguk University) ;
  • RATHIE, ARJUN K. (Department of Mathematics, School of Mathematical and Physical Sciences, Central University of Kerala) ;
  • PRAKASH, OM (Department of Mathematics, Indian Institute of Technology Patna)
  • Received : 2015.04.08
  • Accepted : 2015.07.24
  • Published : 2015.09.25

Abstract

We aim at giving explicit expressions of $${\sum_{m,n=0}^{{\infty}}}{\frac{{\Delta}_{m+n}(-1)^nx^{m+n}}{({\rho})_m({\rho}+i)_nm!n!}$$, where i = 0, ${\pm}1$, ${\ldots}$, ${\pm}9$ and $\{{\Delta}_n\}$ is a bounded sequence of complex numbers. The main result is derived with the help of the generalized Kummer's summation theorem for the series $_2F_1$ obtained earlier by Choi. Further some special cases of the main result considered here are shown to include the results obtained earlier by Kim and Rathie and the identity due to Bailey.

Keywords

Gamma function;Pochhammer symbol;Hypergeometric function;Generalized hypergeometric function;Generalized Kummer's summation theorem

References

  1. W. N. Bailey, Product of generalized hypergeometric series, Proc. London Math. Soc. 228 (1928), 242-254.
  2. W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge (1935); Reprinted by Stechert Hafner, New York (1964).
  3. J. Choi, Contiguous extensions of Dixon's theorem on the sum of a $_3F_2$, J. Inequal. Appl. 2010 (2010), Article ID 589618.
  4. Y. S. Kim and A. K. Rathie, Application of generalized Kummer's summation theorem for the series $_2F_1$, Bull. Koreon Math. Soc. 46(6) (2009), 1201-1211. https://doi.org/10.4134/BKMS.2009.46.6.1201
  5. J. L. Lavoie, F. Grondin, and A. K. Rathie, Generalization of Whipple's theorem on the sum of a $_3F_2$, J. Comput. Appl. Math. 72(2) (1996), 293-300. https://doi.org/10.1016/0377-0427(95)00279-0
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