|
SIGMA 5 (2009), 041, 14 pages arXiv:0904.0561
https://doi.org/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.
pdf (262 kb)
ps (183 kb)
tex (14 kb)
References
- 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.
- 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.
- Aomoto K., Ito M.,
Structure of Jackson integrals of BCn type, Tokyo J. Math. 31 (2008), 449-477.
- Aomoto K., Ito M.,
BCn-type Jackson integral generalized from Gustafson's Cn-type sum,
J. Difference Equ. Appl. 14 (2008), 1059-1097.
- Aomoto K., Ito M.,
A determinant formula for
a holonomic q-difference system
associated with Jackson integrals of type BCn,
Adv. Math., to appear, https://doi.org/10.1016/j.aim.2009.02.003.
- van Diejen J.F., Spiridonov V.P., Modular hypergeometric residue sums of elliptic Selberg integrals, Lett. Math. Phys. 58 (2001), 223-238.
- 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.
- Gasper G., Rahman M.,
Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, Vol. 96, Cambridge University Press, Cambridge, 2004.
- 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.
- 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.
- Ito M.,
q-difference shift for a BCn type Jackson integral arising from 'elementary' symmetric polynomials,
Adv. Math. 204 (2006), 619-646.
- Ito M.,
Another proof of Gustafson's Cn-type summation formula
via 'elementary' symmetric polynomials,
Publ. Res. Inst. Math. Sci. 42 (2006), 523-549.
- 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.
- 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.
- Ito M., Sanada Y.,
On the Sears-Slater basic hypergeometric transformations, Ramanujan J. 17 (2008), 245-257.
- 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.
- Rains E.M., Spiridonov V.P.,
Determinants of elliptic hypergeometric integrals, Funct. Anal. Appl., to appear, arXiv:0712.4253.
|
|