DOI QR코드

DOI QR Code

A NEW PROOF OF THE EXTENDED SAALSCHÜTZ'S SUMMATION THEOREM FOR THE SERIES 4F3 AND ITS APPLICATIONS

Choi, Junesang;Rathie, Arjun K.;Chopra, Purnima

  • Received : 2013.05.23
  • Accepted : 2013.06.03
  • Published : 2013.09.25

Abstract

Very recently, Rakha and Rathie obtained an extension of the classical Saalsch$\ddot{u}$tz's summation theorem. Here, in this paper, we first give an elementary proof of the extended Saalsch$\ddot{u}$tz's summation theorem. By employing it, we next present certain extenstions of Ramanujan's result and another result involving hypergeometric series. The results presented in this paper are simple, interesting and (potentially) useful.

Keywords

Hypergeometric series $_2F_1$;Generalized hypergeometric series $_pF_q$;Saalsch$\ddot{u}$tz's summation theorem and its extension;Ramanujan's formulas

References

  1. M. Abramowitz, I. A. Stegun (Editors), Handbook of Mathematical Functions with Formulas; Graphs; and Mathematical Tables, Applied Mathematics Series 55, ninth printing, National Bureau of Standards, Washington, D.C., 1972.
  2. W. N. Bailey, Products of generalized hypergeometric series, Proc. London Math. Soc. 28(1) (1928), 242-250.
  3. B. C. Berndt, Ramanujan's Notebooks, Part II, Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, and Tokyo, 1989.
  4. J. Choi, Notes on formal manipulations of double series, Commun. Korean Math. Soc. 18(4) (2003), 781-789. https://doi.org/10.4134/CKMS.2003.18.4.781
  5. P. Chopra and A. K. Rathie, A result closely related to the Ramanujan's result, submitted for publication, 2013.
  6. Y. S. Kim and A. K. Rathie, Comment on 'A summation formula for Clausen's series $_3F_2$(1) with an application to Goursat's function $_2F_2$(x)', J. Phys. A: Math. Theor. 41 (2008) 078001 (2pp). https://doi.org/10.1088/1751-8113/41/7/078001
  7. A. R. Miller, A summation formula for Clausen's series $_3F_2$(1) with an application to Goursat's function $_2F_2$(x), J. Phys. A: Math. General 16 (2005), 3541-3545.
  8. A. P. Prudnikov, Yu. A. Brychkov and O. I. Marchev, Integrals and Series, More Special Functions, 3, Gordon and Breach Science Publishers, 1990; Translated from the Russian Edition in 1986 by G. G. Gould.
  9. E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
  10. A. K. Rathie and R. Paris, An extension of the Euler-type transformation for the $_3F_2$ series, Far East J. Math. Sci. (FJMS), 27(1) (2007), 43-48.
  11. M. A. Rakha and A. K. Rathie, Extensions of Euler's type II transformation and Saalschutz's theorem, Bull. Korean Math. Soc. 48(1) (2011), 151-156. https://doi.org/10.4134/BKMS.2011.48.1.151
  12. H. M. Srivastava, J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London, and New York, 2012.

Cited by

  1. An alternative proof of the extended Saalschütz summation theorem for ther+ 3Fr+ 2(1) series with applications vol.38, pp.18, 2015, https://doi.org/10.1002/mma.3408

Acknowledgement

Supported by : National Research Foundation of Korea