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\(U(n+1)\) WP-Bailey tree

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Abstract

The purpose of this paper is to establish a \(U(n+1)\) WP-Bailey tree. As an application, we use a suitable \(U(n+1)\) WP-Bailey pair to recover one of the \(U(n+1)\) \(_{10}\phi _9\) transformation formulas established by Milne and Newcomb.

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References

  1. Andrews, G.E.: Bailey’s transform, lemma, chains and tree. In: Bustoz, J., et al. (eds.) Special Functions 2000: Current Perspective and Future Directions, pp. 1–22. Kluwer Academic Publishers, Dordrecht (2001)

    Chapter  Google Scholar 

  2. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  3. Andrews, G.E., Berkovich, A.: The \(WP\)-Bailey tree and its implications. J. Lond. Math. Soc. 66(2), 529–549 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bressoud, D.: Some identities for terminating \(q\)-series. Math. Proc. Camb. Philos. Soc. 89(2), 211–223 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications, vol. 96, 2nd edn. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  6. Jouhet, F.: Shifted versions of the Bailey and well-poised Bailey lemmas. Ramanujan J. 23, 315–333 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Mc Laughlin, J., Zimmer, P.: Some identities between basic hypergeometric series deriving from a new Bailey-type transformation. J. Math. Anal. Appl. 345, 670–677 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mc Laughlin, J., Zimmer, P.: Some implications of the \(WP\)-Bailey tree. Adv. Appl. Math. 43, 162–175 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mc Laughlin, J., Zimmer, P.: General \(WP\)-Bailey chains. Ramanujan J. 22(1), 11–31 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Mc Laughlin, J., Zimmer, P.: A reciprocity relation for \(WP\)-Bailey pairs. Ramanujan J. 28, 155–173 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mc Laughlin, J., Sills, A.V., Zimmer, P.: Lifting Bailey pairs to \(WP\)-Bailey pairs. Discret. Math. 309, 5077–5091 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lilly, G.M., Milne, S.C.: The \(A_l\) and \(C_l\) Bailey transform and lemma. Bull. Am. Math. Soc. 26, 258–263 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lilly, G.M., Milne, S.C.: The \(C_l\) Bailey transform and Bailey lemma. Constr. Approx. 9, 473–500 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, Q., Ma, X.: On a characteristic equation of well-poised Bailey chains. Ramanujan J. 18, 351–370 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lovejoy, J.: A Bailey lattice. Proc. Am. Math. Soc. 132(5), 1507–1516 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Milne, S.C.: Multiple \(q\)-series and \(U\!(n)\) generalizations of Ramanujan’s \(_1\psi _1\) sum. In: Andrews, G.E., et al. (eds.) Ramanujan Revisited, pp. 473–524. Academic Press, New York (1988)

    Google Scholar 

  17. Milne, S.C.: The multidimensional \(_1\Psi _1\) sum and Macdonald identities for \(A_\ell ^{(1)}\). In: Ehrenpreis, L., Gunning, R.C. (eds.) Theta Functions Bowdoin 1987. Proceedings of Symposia in Pure Mathematics, vol. 49 (Part 2), pp. 323–359 (1989)

  18. Milne, S.C.: Balanced \(_3\phi _2\) summation theorems for \(U(n)\) basic hypergeometric series. Adv. Math. 131, 93–187 (1997)

    Article  MathSciNet  Google Scholar 

  19. Milne, S.C., Lilly, G.M.: Consequences of the \(A_l\) and \(C_l\) Bailey transform and Bailey lemma. Discret. Math. 139, 319–346 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Milne, S.C., Newcomb, J.W.: \(U(n)\) very-well-poised \(_{10}\phi _9\) transformations. J. Comput. Appl. Math. 68, 239–285 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rowell, M.J.: A new general conjugate Bailey pair. Pac. J. Math. 238(2), 367–385 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Singh, U.S.: A note on a transformation of Bailey. Q. J. Math. Oxf. Ser. 2 45(177), 111–116 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Spiridonov, V.P.: An elliptic incarnation of the Bailey chain. Int. Math. Res. Not. 37, 1945–1977 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Spiridonov, V.P.: A Bailey tree for integrals. Theor. Math. Phys. 139, 536–541 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Warnaar, S.O.: 50 Years of Bailey’s lemma. In: Betten, A., et al. (eds.) Algebraic Combinatorics and Applications. Springer, Berlin (2001)

    Google Scholar 

  26. Warnaar, S.O.: Extensions of the well-poised and elliptic well-poised Bailey lemma. Indag. Math. 14, 571–588 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors are very grateful to anonymous referees for their comments and valuable suggestions which helped improve the original version of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Luoyang Normal University, Luoyang, 471022, People’s Republic of China

    Zhizheng Zhang

  2. College of Mathematics and Information Science, Henan Normal University, Xinxiang, 475001, People’s Republic of China

    Qiyan Liu

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  1. Zhizheng Zhang
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  2. Qiyan Liu
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Corresponding author

Correspondence to Zhizheng Zhang.

Additional information

This research was supported by the National Natural Science Foundation of China (Grant No. 11371184).

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Zhang, Z., Liu, Q. \(U(n+1)\) WP-Bailey tree. Ramanujan J 40, 447–462 (2016). https://doi.org/10.1007/s11139-016-9790-4

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  • DOI: https://doi.org/10.1007/s11139-016-9790-4

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