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Watson–Like Formulae for Terminating 3 F 2-Series

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Advances in Combinatorics

Abstract

Several closed formulae are established for terminating Watson–like hypergeometric 3 F 2-series by investigating, through Gould and Hsu’s fundamental pair of inverse series relations, the dual relations of Dougall’s formula for the very well–poised 5 F 4-series.

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References

  1. W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935.

    MATH  Google Scholar 

  2. W. Chu, Inversion techniques and combinatorial identities, Boll. Un. Mat. Ital. 7-B (1993), 737–760.

    Google Scholar 

  3. W. Chu, Inversion techniques and combinatorial identities: strange evaluations of hypergeometric series, Pure Math. Appl. 4:4 (1993), 409–428.

    MathSciNet  MATH  Google Scholar 

  4. W. Chu, Inversion techniques and combinatorial identities: a quick introduction to hypergeometric evaluations, Runs and patterns in probability (Kluwer Acad. Publ., Dordrecht, 1994): Math. Appl. 283 (1994), 31–57.

    Google Scholar 

  5. W. Chu, Inversion techniques and combinatorial identities: A unified treatment for the 7 F 6 -series identities, Collectanea Mathematica 45:1 (1994), 13–43.

    MathSciNet  MATH  Google Scholar 

  6. W. Chu, Inversion techniques and combinatorial identities: balanced hypergeometric series, Rocky Mountain J. Math. 32:2 (2002), 561–587.

    Article  MathSciNet  MATH  Google Scholar 

  7. W. Chu, Analytical formulae for extended 3 F 2 -series of Watson–Whipple-Dixon with two extra integer parameters, Mathematics of Computation 81:277 (2012), 461–479.

    Article  Google Scholar 

  8. J. Dougall, On Vandermonde’s theorem and some more general expansions, Proc. Edinburgh Math. Soc. 25 (1907), 114–132.

    Article  MATH  Google Scholar 

  9. H. W. Gould, L. C. Hsu, Some new inverse series relations, Duke Math. J. 40 (1973), 885–891.

    Article  MathSciNet  MATH  Google Scholar 

  10. I. M. Gessel, P. J. Larcombe, The sum \(1{6}^{n}\sum _{k=0}^{2n}{4}^{k}\left ({ \frac{1} {2} \atop k} \right )\left ({ -\frac{1} {2} \atop k} \right )\left ({ -2k \atop 2n-k} \right )\) : a third proof of its closed form, Utilitas Math. 80 (2009), 59–63.

    MathSciNet  MATH  Google Scholar 

  11. W. A. Koepf, P. J. Larcombe, The sum \(1{6}^{n}\sum _{k=0}^{2n}{4}^{k}\left ({ \frac{1} {2} \atop k} \right )\left ({ -\frac{1} {2} \atop k} \right )\left ({ -2k \atop 2n-k} \right )\) : a computer assisted proof of its closed form and some generalised results, Utilitas Math. 79 (2009), 9–15.

    MathSciNet  MATH  Google Scholar 

  12. P. J. Larcombe, M. E. Larsen, The sum \(1{6}^{n}\sum _{k=0}^{2n}{4}^{k}\left ({ \frac{1} {2} \atop k} \right )\left ({ -\frac{1} {2} \atop k} \right )\left ({ -2k \atop 2n-k} \right )\) : A proof of its closed form, Utilitas Math. 79 (2009), 3–7.

    MathSciNet  MATH  Google Scholar 

  13. S. Lewanowicz, Generalized Watson’s summation formula for 3 F 2 (1), J. Comput. Appl. Math. 86 (1997), 375–386.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. K. Rathie, R. B. Paris, A New Proof of Watson’s Theorem for the Series 3 F 2 (1), Appl. Math. Sci. 3:4 (2009), 161–164.

    MathSciNet  MATH  Google Scholar 

  15. G. N. Watson, A note on generalized hypergeometric series, Proc. London Math. Soc. (2) 23 (1925), xiii–xv.

    Google Scholar 

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Correspondence to Wenchang Chu .

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Chu, W., Zhou, R.R. (2013). Watson–Like Formulae for Terminating 3 F 2-Series. In: Kotsireas, I., Zima, E. (eds) Advances in Combinatorics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30979-3_7

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