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New transformations for elliptic hypergeometric series on the root system A n

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Abstract

Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system A n , with different dimensions n. We give, with a new, elementary proof, an elliptic extension of this transformation. We also obtain further Bailey-type transformations as consequences of our result, some of which are new also in the case of basic and classical hypergeometric series.

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References

  1. Askey, R.: Some basic hypergeometric extensions of integrals of Selberg and Andrews. SIAM J. Math. Anal. 11, 938–951 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bailey, W.N.: An identity involving Heine’s basic hypergeometric series. J. London Math. Soc. 4, 254–257 (1929)

    Google Scholar 

  3. Denis, R.Y., Gustafson, R.A.: An SU(n) q-beta integral transformation and multiple hypergeometric series identities. SIAM J. Math. Anal. 23, 552–561 (1992)

  4. van Diejen, J.F., Spiridonov, V.P.: An elliptic Macdonald-Morris conjecture and multiple modular hypergeometric sums. Math. Res. Lett. 7, 729–746 (2000)

    MATH  MathSciNet  Google Scholar 

  5. van Diejen, J.F., Spiridonov, V.P.: Modular hypergeometric residue sums of elliptic Selberg integrals. Lett. Math. Phys. 58, 223–238 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Frenkel, I.B., Turaev, V.G.: Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions. In The Arnold–Gelfand Mathematical Seminars (V.I. Arnold et al., eds.), Birkhäuser, Boston, pp. 171–204 (1997)

    Google Scholar 

  7. Gasper, G., Rahman, M.: Basic hypergeometric series. Cambridge University Press, Cambridge (1990)

  8. Gessel, I.M., Krattenthaler, C.: Cylindric partitions. Trans. Amer. Math. Soc. 349, 429–479 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gustafson, R.A.: Multilateral summation theorems for ordinary and basic hypergeometric series in U(n). SIAM J. Math. Anal. 18, 1576–1596 (1987)

    Google Scholar 

  10. Holman, W.J., Biedenharn, L.C., Louck, J.D.: On hypergeometric series well-poised in SU(n). SIAM J. Math. Anal. 7, 529–541 (1976)

    Google Scholar 

  11. Kajihara, Y.: Some remarks on multiple Sears transformations. in q-Series with Applications to Combinatorics, Number Theory, and Physics (B.C. Berndt and K. Ono, eds.), Contemp. Math. 291, Amer. Math. Soc., Providence, pp. 139–145 (2001)

  12. Kajihara, Y.: Euler transformation formulas for multiple basic hypergeometric series of type A and some applications. Adv. Math. 187, 53–97 (2004)

    Google Scholar 

  13. Kajihara, Y., Noumi, M.: Multiple elliptic hypergeometric series – An approach from the Cauchy determinant. Indag. Math. 14, 395–421 (2003)

    Google Scholar 

  14. Kaneko, J.: Constant term identities of Forrester–Zeilberger–Cooper. Discrete Math. 173, 79–90 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Krattenthaler, C.: Proof of a summation formula for an à n basic hypergeometric series conjectured by Warnaar. In q-Series with Applications to Combinatorics, Number Theory, and Physics (B.C. Berndt and K. Ono, eds.), Contemp. Math. 291, Amer. Math. Soc., Providence, pp. 153–161 (2001)

  16. Milne, S.C.: Multiple q-series and U(n) generalizations of Ramanujan’s 1Ψ1 sum. In Ramanujan Revisited (G.E. Andrews et al., eds.), Academic Press, Boston, pp. 473–524 (1988)

    Google Scholar 

  17. Milne, S.C., Newcomb, J.W.: U(n) very-well-poised 10φ9 transformations. J. Comput. Appl. Math. 68, 239–285 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Rains, E.M.: Transformations of elliptic hypergometric integrals”, e-print, 2003, math.QA/0309252

  19. Rosengren, H.: A proof of a multivariable elliptic summation formula conjectured by Warnaar. in q-Series with Applications to Combinatorics, Number Theory, and Physics (B.C. Berndt and K. Ono, eds.), Contemp. Math. 291, Amer. Math. Soc., Providence, pp. 193–202 (2001)

  20. Rosengren, H.: Karlsson–Minton type hypergeometric functions on the root system C n . J. Math. Anal. Appl. 281, 332–345 (2003)

    MATH  MathSciNet  Google Scholar 

  21. Rosengren, H.: Reduction formulas for Karlsson–Minton-type hypergeometric functions. Constr. Approx. 20, 525–548 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Rosengren, H.: Elliptic hypergeometric series on root systems. Adv. Math. 181, 417–447 (2004)

    Google Scholar 

  23. Rosengren, H., Schlosser, M.: Summations and transformations for multiple basic and elliptic hypergeometric series by determinant evaluations. Indag. Math. 14, 483–513 (2003)

    Google Scholar 

  24. Spiridonov, V.P.: Theta hypergeometric series. In Asymptotic Combinatorics with Applications to Mathematical Physics (V.A. Malyshev and A.M. Vershik, eds.), Kluwer Acad. Publ., Dordrecht, pp. 307–327 (2002)

  25. Spiridonov, V.P.: Theta hypergeometric integrals. St. Petersburg Math. J. 15, 929–967 (2004)

    Article  MathSciNet  Google Scholar 

  26. Tarasov, V. Varchenko, A.: Identities for hypergeometric integrals of different dimensions. Lett. Math. Phys. 71, 89–99 (2005)

    Google Scholar 

  27. Warnaar, S.O.: Summation and transformation formulas for elliptic hypergeometric series. Constr. Approx. 18, 479–502 (2002)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Chalmers University of Technology and Göteborg University, SE-412 96, Göteborg, Sweden

    Hjalmar Rosengren

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  1. Hjalmar Rosengren
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Correspondence to Hjalmar Rosengren.

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2000 Mathematics Subject Classification Primary—33D67; Secondary—11F50

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Rosengren, H. New transformations for elliptic hypergeometric series on the root system A n . Ramanujan J 12, 155–166 (2006). https://doi.org/10.1007/s11139-006-0070-6

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  • DOI: https://doi.org/10.1007/s11139-006-0070-6

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