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ON BASIC ANALOGUE OF CLASSICAL SUMMATION THEOREMS DUE TO ANDREWS

Harsh, Harsh Vardhan;Rathie, Arjun K.;Purohit, Sunil Dutt

  • Received : 2015.05.14
  • Accepted : 2015.12.23
  • Published : 2016.03.25

Abstract

In 1972, Andrews derived the basic analogue of Gauss's second summation theorem and Bailey's theorem by implementing basic analogue of Kummer's theorem into identity due to Jackson. Recently Lavoie et.al. derived many results closely related to Kummer's theorem, Gauss's second summation theorem and Bailey's theorem and also Rakha et. al. derive the basic analogues of results closely related Kummer's theorem. The aim of this paper is to derive basic analogues of results closely related Gauss's second summation theorem and Bailey's theorem. Some applications and limiting cases are also considered.

Keywords

Basic hypergeometric function;generalized Gauss's second summation theorem;generalized Bailey's theorem;Kummer's theorem;Jackson's identity

References

  1. G. E. Andrews, On the q-anlog of Kummer's theorem and applications, Duke Math. J. 40 (1973), 525-528. https://doi.org/10.1215/S0012-7094-73-04045-3
  2. W.N. Bailey, A note on certain q-identities, Quart. J. Math. 12 (1941), 173-175.
  3. J.A. Dhaum, The basic analog of Kummer's theorem, Bull. Amer. Math. Soc. 48 (1942), 711-713. https://doi.org/10.1090/S0002-9904-1942-07764-0
  4. F. H. Jackson, Transformations of q-series, Mess. Muth. 39 (1910), 145-153.
  5. G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge and New York, 35 (1990).
  6. E. Heine, Handbuch der Kugelfunctionen,Theorie und Anwendungen, Reimer, Berlin, Vol.1 (1878)
  7. J.L. Lavoie, F. Grondin, A.K. Rathie and K. Arora, Generalizations of Dixon's theorem on the sum of a $_3F_2$, Computation 62 (1994), 267-276.
  8. J.L. Lavoie, F. Grondin and A.K. Rathie, Generalizations of Watson's theorem on the sum of $_3F_2$, Indian J. Math. 34 (1992), 23-32.
  9. J.L. Lavoie, F. Grondin and A.K. Rathie, Generalizations of Whipple's theorem on the sum of $_3F_2$, J. Comput. Appl. Math. 72 (1996), 293-300. https://doi.org/10.1016/0377-0427(95)00279-0
  10. M.A. Rakha, Y.S. Kim, A.K. Rathie and H.V. Harsh, A study of q-contiguous functions relations, Comm. Korean Math. Soc., Accepted.