On Elliptic Product Formulas for Jackson Integrals Associated with Reduced Root Systems
Kazuhiko Aomoto
DOI: 10.1023/A:1008629309210
Abstract
In this note, we state certain product formulae for Jackson integrals associated with any root systems, involved in elliptic theta functions which appear as connection coefficients. The fomulae arise naturally in case of arbitrary root systems by extending the connection problem which has been investigated in [1, 4] in case of A type root system. This is also connected with the Macdonald-Morris constant term identity investigated by I. Cherednik [6], and K. Kadell [15] on the one hand, and of the Askey-Habsieger-Kadell”s q-Selberg integral formula and its extensions [4, 8, 12, 14, 15] on the other. This is also related with some of the results due to R.A. Gustafson [10, 11], although our integrands are different from his.
Pages: 115–126
Keywords: elliptic theta function; Jackson integral; reduced root system; product formula; $q$-difference
Full Text: PDF
References
1. K. Aomoto, “On connection coefficients for q-difference system of A type Jackson integral,” SIAM J. Math. Anal. 25 (1994), 256-273.
2. K. Aomoto, “On a theta product formula for the symmetric A type connection function,” Osaka J. Math. 32 (1995), 35-39.
3. K. Aomoto and Y. Kato, “Gauss decomposition of connection matrices and application to Yang-Baxter equation, I,” Proc. Japan Acad. 69 (1993), 238-241; II, ibid., 341-344.
4. K. Aomoto and Y. Kato, “Connection coefficients for symmetric A type Jackson integrals,” Duke Math. Jour. 74 (1994), 129-143.
5. R. Askey, “Some basic hypergeometric extensions of integrals of Selberg and Andrews,” SIAM J. Math. Anal. 11 (1980), 938-951.
6. I. Cherednik, “The Macdonald constant term conjecture,” Int. Math. Res. Notices (6) (1993), 165-177.
7. J.F. van Diejen, “On certain multiple Bailey, Rogers and Dougall type summation formulas,” Pub. of R.I.M.S., 33(1997), 483-508.
8. R. Evans, “Multidimensional q-Beta integrals,” SIAM J. Math. Anal. 23 (1992), 758-765.
9. P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley and Sons, 1978.
10. R.A. Gustafson, “Some q-Beta integrals on SU (n) and Sp(n) that generalize the Askey-Wilson and Nasrallah- Rahman integrals,” SIAM J. Math. Anal. 25 (1994), 441-449.
11. R.A. Gustafson, “Some q-beta and Mellin-Barnes integrals on compact Lie groups and Lie algebras,” Trans. AMS 341 (1994), 69-119.
12. L. Habsieger, “Une q-integral de Selberg et Askey,” Trans. AMS 19 (1988), 1475-1489.
13. Masa Ito, “On theta product formula for Jackson integrals associated with root systems of rank two,” preprint, 1995.
14. K. Kadell, “A proof of Askey's conjectured q-analog of Selberg's integral and a conjecture of Morris,” SIAM J. Math. Anal. 19 (1988), 969-986.
15. K. Kadell, “A proof of the q Macdonald-Morris conjecture for BCn,” Mem. AMS 108 (516) (1994), 1-80.
16. J. Kaneko, “q-Selberg integrals and Macdonald polynomials,” Ann. Ecole Norm. Sup. 29 (1996), 583-637.
17. I. Macdonald, “A formal identity for affine root systems,” preprint, 1996.
18. K. Mimachi, “Connection problem in holonomic q-difference system associated with a Jackson integral of Jordan-Pochhammer type,” Nagoya Math. J. 116 (1989), 149-161.
19. T. Terasoma, “Determinants of q-hypergeometric functions and another proof of Askey conjecture,” preprint, 1995.
2. K. Aomoto, “On a theta product formula for the symmetric A type connection function,” Osaka J. Math. 32 (1995), 35-39.
3. K. Aomoto and Y. Kato, “Gauss decomposition of connection matrices and application to Yang-Baxter equation, I,” Proc. Japan Acad. 69 (1993), 238-241; II, ibid., 341-344.
4. K. Aomoto and Y. Kato, “Connection coefficients for symmetric A type Jackson integrals,” Duke Math. Jour. 74 (1994), 129-143.
5. R. Askey, “Some basic hypergeometric extensions of integrals of Selberg and Andrews,” SIAM J. Math. Anal. 11 (1980), 938-951.
6. I. Cherednik, “The Macdonald constant term conjecture,” Int. Math. Res. Notices (6) (1993), 165-177.
7. J.F. van Diejen, “On certain multiple Bailey, Rogers and Dougall type summation formulas,” Pub. of R.I.M.S., 33(1997), 483-508.
8. R. Evans, “Multidimensional q-Beta integrals,” SIAM J. Math. Anal. 23 (1992), 758-765.
9. P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley and Sons, 1978.
10. R.A. Gustafson, “Some q-Beta integrals on SU (n) and Sp(n) that generalize the Askey-Wilson and Nasrallah- Rahman integrals,” SIAM J. Math. Anal. 25 (1994), 441-449.
11. R.A. Gustafson, “Some q-beta and Mellin-Barnes integrals on compact Lie groups and Lie algebras,” Trans. AMS 341 (1994), 69-119.
12. L. Habsieger, “Une q-integral de Selberg et Askey,” Trans. AMS 19 (1988), 1475-1489.
13. Masa Ito, “On theta product formula for Jackson integrals associated with root systems of rank two,” preprint, 1995.
14. K. Kadell, “A proof of Askey's conjectured q-analog of Selberg's integral and a conjecture of Morris,” SIAM J. Math. Anal. 19 (1988), 969-986.
15. K. Kadell, “A proof of the q Macdonald-Morris conjecture for BCn,” Mem. AMS 108 (516) (1994), 1-80.
16. J. Kaneko, “q-Selberg integrals and Macdonald polynomials,” Ann. Ecole Norm. Sup. 29 (1996), 583-637.
17. I. Macdonald, “A formal identity for affine root systems,” preprint, 1996.
18. K. Mimachi, “Connection problem in holonomic q-difference system associated with a Jackson integral of Jordan-Pochhammer type,” Nagoya Math. J. 116 (1989), 149-161.
19. T. Terasoma, “Determinants of q-hypergeometric functions and another proof of Askey conjecture,” preprint, 1995.