Honam Mathematical Journal (호남수학학술지)

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NOTE ON THE CLASSICAL WATSON'S THEOREM FOR THE SERIES 3F2

  • Choi, Junesang (Department of Mathematics, Dongguk University) ;
  • Agarwal, P. (Department of Mathematics, Anand International College of Engineering)
  • Received : 2013.09.16
  • Accepted : 2013.10.15
  • Published : 2013.12.25
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Abstract

Summation theorems for hypergeometric series $_2F_1$ and generalized hypergeometric series $_pF_q$ play important roles in themselves and their diverse applications. Some summation theorems for $_2F_1$ and $_pF_q$ have been established in several or many ways. Here we give a proof of Watson's classical summation theorem for the series $_3F_2$(1) by following the same lines used by Rakha [7] except for the last step in which we applied an integral formula introduced by Choi et al. [3].

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References

  1. W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, New York, 1964.
  2. R. C. Bhatt, Another proof of Watson's theorem for summing $_{3}F_{2}$(1), J. London Math. Soc. 40 (1965), 47-48.
  3. J. Choi, A. K. Rathie and Purnima, A Note on Gauss's Second Summation Theorem for the Series $_{2}F_{1}$ ($\frac{1}{2}$), Commun. Korean Math. Soc. 22(4) (2007), 509-512. https://doi.org/10.4134/CKMS.2007.22.4.509
  4. A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol. II, McGraw-Hill Book Company, New York, Toronto and London, 1954.
  5. T. M. MacRobert, Functions of Complex Variables, 5th edition, Macmillan, London, 1962.
  6. E. D. Rainville, Special Functions, Macmillan, New York, 1960.
  7. M. A. Rakha, A new proof of the classical Watson's summation theorem, Appl. Math. E-Notes 11 (2011), 278-282.
  8. A. K. Rathie and R. B. Paris, A new proof of Watson's theorem for the series $_{3}F_{2}$(1), App. Math. Sci. 3(4) (2009), 161-164.
  9. M. Raussen and C. Skau, Interview with Michael Atiya and Isadore Singer, Notices Amer. Math. Soc. 52 (2005), 225-233.
  10. H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London, and New York (2012).
  11. G. N. Watson, A note on generalized hypergeometric series, Proc. London Math. Soc. 2(23) (1925), 13-15.
  12. F. J. Whipple, A group of generalized hypergeometric series; relations between 120 allied series of type F(a, b, c; e, f), Proc. London Math. Soc. 2(23) (1925), 104-114.