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Reduction and summation formulae for semi-terminating q-Kampé de Fériet series

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Abstract

Three classes of semi-terminating q-Kampé de Fériet series are systematically investigated via the transformations of Sears and Watson on the terminating balanced \({_4\phi_3}\) -series. Thirteen transformation theorems are established, which are utilized to derive several reduction and summation formulae for bivariate basic hypergeometric series.

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References

  1. Al-Salam W.A.: Saalschütz for basic double series. J. Lond. Math. Soc. 40, 455–458 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bailey W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935)

    MATH  Google Scholar 

  3. Carlitz L.: A Saalschützian theorem for double series. J. Lond. Math. Soc. 38, 415–418 (1963)

    Article  MATH  Google Scholar 

  4. Carlitz L.: Summation of a double hypergeometric series. Matematiche (Catania) 22, 138–142 (1967)

    MathSciNet  MATH  Google Scholar 

  5. Chu W., Jia C.: Bivariate classical and q-series transformations. Port. Math. 65(2), 243–256 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chu W., Jia C.: Transformation and reduction formulae for double q-Clausen hypergeometric series. Math. Methods Appl. Sci. 31:1, 1–17 (2008)

    Article  MathSciNet  Google Scholar 

  7. Chu W., Srivastava H.M.: Ordinary and basic bivariate hypergeometric transformations associated with the Appell and Kampé de Fériet functions. J. Comput. Appl. Math. 156(2), 355–370 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gasper G., Rahman M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  9. Jain V.K.: Some expansions involving basic hypergeometric functions of two variables. Pac. J. Math. 91(2), 349–361 (1980)

    Article  MATH  Google Scholar 

  10. Jia C., Wang T.: Transformation and reduction formulae for double q-Clausen series of type \({\Phi^{1:2;\lambda}_{1:1;\mu} }\) . J. Math. Anal. Appl. 328(1), 609–624 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jia C., Wang T.: Reduction and transformation formulae for bivariate basic hypergeometric series. J. Math. Anal. Appl. 328(2), 1152–1160 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jia C., Zhang X.: Transformation and reduction formulae for double q-series of type \({\Phi^{2:1;\lambda}_{2:0;\mu} }\) . Glasgow Math. J. 52, 195–204 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Karlsson P.W.: Some reducible generalized Kampé de Fériet functions. J. Math. Anal. Appl. 96, 546–550 (1983)

    Article  MathSciNet  Google Scholar 

  14. Karlsson P.W.: Two hypergeometric summation formulae related to 9−j coefficients. J. Phys. A: Math. Gen. 27, 6943–6945 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Karlsson, P.W.: Reduction of double Clauseian functions. Int. J. Math. Statist. Sci. 5, 33–49 (1996); corrigendum 6, 235 (1997)

    Google Scholar 

  16. Karlsson P.W.: Some formulae for double Clauseian functions. J. Comput. Appl. Math. 118, 203–213 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lievens S., Van der Jeugt J.: Transformation formulas for double hypergeometric series related to 9−j coefficients and their basic analogs. J. Math. Phys. 41:11, 5417–5430 (2001)

    Article  MathSciNet  Google Scholar 

  18. Pitre S.N., Van der Jeugt J.: Transformation and summation formulas for Kampé de Fériet series \({F^{0:3}_{1:1} }\) (1,1). J. Math. Anal. Appl. 202, 121–132 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Singh S.P.: Certain transformation formulae involving basic hypergeometric functions. J. Math. Phys. Sci. 28(4), 189–195 (1994)

    MathSciNet  MATH  Google Scholar 

  20. Sighal R.P.: Transformation formulae for the modified Kampé de Fériet function. The Math. Student 39, 189–195 (1972)

    Google Scholar 

  21. Sighal R.P.: Certain transformation formulae involving basic hypergeometric functions. J. Math. Phys. Sci. 38, 189–195 (1994)

    Google Scholar 

  22. Srivastava H.M., Jain V.K.: q-Series identities and reducibility of basic double hypergeometric functions. Canad. J. Math. 38:1, 215–231 (1986)

    Article  MathSciNet  Google Scholar 

  23. Srivastava H.M., Karlsson P.W.: Multiple Gaussian Hypergeometric Series. Halsted Press, New York (1985)

    MATH  Google Scholar 

  24. Vander Jeugt J.: Transformation formula for a double Clausenian hypergeometric series, its q-analogue, and its invariance group. J. Comput. Appl. Math. 139(1), 65–73 (2002)

    Article  MathSciNet  Google Scholar 

  25. Van der Jeugt J., Pitre S.N., Srinivasa Rao K.: Transformation and summation formulas for double hypergeometric series. J. Comput. Appl. Math. 83, 185–193 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Wenchang Chu & Nadia N. Li

  2. Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Universitá del Salento, Lecce–Arnesano, P.O. Box 193, 73100, Lecce, Italy

    Wenchang Chu

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  1. Wenchang Chu
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  2. Nadia N. Li
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Correspondence to Wenchang Chu.

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Chu, W., Li, N.N. Reduction and summation formulae for semi-terminating q-Kampé de Fériet series. Afr. Mat. 24, 647–664 (2013). https://doi.org/10.1007/s13370-012-0085-7

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  • DOI: https://doi.org/10.1007/s13370-012-0085-7

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