Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 041, 14 pages      arXiv:0904.0561      doi:10.3842/SIGMA.2009.041
Contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions”

A First Order q-Difference System for the BC1-Type Jackson Integral and Its Applications

Masahiko Ito
Department of Physics and Mathematics, Aoyama Gakuin University, Kanagawa 229-8558, Japan

Received December 01, 2008, in final form March 18, 2009; Published online April 03, 2009

Abstract
We present an explicit expression for the q-difference system, which the BC1-type Jackson integral (q-series) satisfies, as first order simultaneous q-difference equations with a concrete basis. As an application, we give a simple proof for the hypergeometric summation formula introduced by Gustafson and the product formula of the q-integral introduced by Nassrallah-Rahman and Gustafson.

Key words: q-difference equations; Jackson integral of type BC1; Gustafson's Cn-type sum; Nassrallah-Rahman integral.

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References

  1. Aomoto K., A normal form of a holonomic q-difference system and its application to BC1-type, Int. J. Pure Appl. Math. 50 (2009), 85-95.
  2. Aomoto K., Ito M., On the structure of Jackson integrals of BCn type and holonomic q-difference equations, Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), 145-150.
  3. Aomoto K., Ito M., Structure of Jackson integrals of BCn type, Tokyo J. Math. 31 (2008), 449-477.
  4. Aomoto K., Ito M., BCn-type Jackson integral generalized from Gustafson's Cn-type sum, J. Difference Equ. Appl. 14 (2008), 1059-1097.
  5. Aomoto K., Ito M., A determinant formula for a holonomic q-difference system associated with Jackson integrals of type BCn, Adv. Math., to appear, doi:10.1016/j.aim.2009.02.003.
  6. van Diejen J.F., Spiridonov V.P., Modular hypergeometric residue sums of elliptic Selberg integrals, Lett. Math. Phys. 58 (2001), 223-238.
  7. Denis R.Y., Gustafson R.A., An SU(n) q-beta integral transformation and multiple hypergeometric series identities, SIAM J. Math. Anal. 23 (1992), 552-561.
  8. Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, Vol. 96, Cambridge University Press, Cambridge, 2004.
  9. Gustafson R.A., Some q-beta and Mellin-Barnes integrals with many parameters associated to the classical groups, SIAM J. Math. Anal. 23 (1992), 525-551.
  10. Gustafson R.A., Some q-beta and Mellin-Barnes integrals on compact Lie groups and Lie algebras, Trans. Amer. Math. Soc. 341 (1994), 69-119.
  11. Ito M., q-difference shift for a BCn type Jackson integral arising from 'elementary' symmetric polynomials, Adv. Math. 204 (2006), 619-646.
  12. Ito M., Another proof of Gustafson's Cn-type summation formula via 'elementary' symmetric polynomials, Publ. Res. Inst. Math. Sci. 42 (2006), 523-549.
  13. Ito M., A multiple generalization of Slater's transformation formula for a very-well-poised-balanced 2rψ2r series, Q. J. Math. 59 (2008), 221-235.
  14. Ito M., Okada S., An application of Cauchy-Sylvester's theorem on compound determinants to a BCn-type Jackson integral, in Proceedings of the Conference on Partitions, q-Series and Modular Forms (University of Florida, March 12-16, 2008), to appear.
  15. Ito M., Sanada Y., On the Sears-Slater basic hypergeometric transformations, Ramanujan J. 17 (2008), 245-257.
  16. Nassrallah B., Rahman M., Projection formulas, a reproducing kernel and a generating function for q-Wilson polynomials, SIAM J. Math. Anal. 16 (1985), 186-197.
  17. Rains E.M., Spiridonov V.P., Determinants of elliptic hypergeometric integrals, Funct. Anal. Appl., to appear, arXiv:0712.4253.

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