On the structure of Jackson integrals of \(BC_n\) type and holonomic \(q\)-difference equations. (English) Zbl 1155.33010

The authors study a particular \(q\)-analogue of de Rham cohomology complex related to so-called Jackson integrals; the latter are actually by definition sums over multiplicative lattices in \((\mathbb{C}^{*})^{n}\). In the case of Jackson integrals of type \(BC_n\), the paper studies separately symmetric and non-symmetric cohomologies, giving explicit bases. Last, it is proved that these integrals satisfy holonomic systems of linear \(q\)-difference equations with respect to the parameters.


33D67 Basic hypergeometric functions associated with root systems
39A12 Discrete version of topics in analysis
Full Text: DOI


[1] K. Aomoto, \(q\)-analogue of de Rham cohomology associated with Jackson integrals I, II, Proc. Japan Acad. 66A (1990), 161-164; 240-244. · Zbl 0718.33012 · doi:10.3792/pjaa.66.240
[2] K. Aomoto, Connection formulas in the \(q\)-analog de Rham cohomology, in Functional analysis on the eve of the 21st century , Vol . 1 ( New Brunswick , NJ , 1993), 1-12, Progr. Math., 131, Birkhäuser Boston, Boston, MA, 1995. · Zbl 0845.33010 · doi:10.1007/978-1-4612-2582-9_1
[3] K. Aomoto, \(q\)-difference de Rham complex and Čech cohomology (an essay on basic hypergeometric functions), Sugaku Expositions, Sugaku Expositions 13 (2000), no. 2, 125-142. · Zbl 0943.32010
[4] K. Aomoto and M. Ito, \(BC_n\) type Jackson integral generalized from Gustafson’s \(C_n\) type sum, (2004). (Preprint). · Zbl 1155.33009 · doi:10.1080/10236190701776043
[5] K. Aomoto and M. Ito, On the Structure of Jackson Integrals of \(BC_n\) type, (2005). (Preprint). · Zbl 1155.33010 · doi:10.3792/pjaa.81.145
[6] K. Aomoto and M. Ito, A determinant formula for a holonomic \(q\)-difference system of \(BC_n\) type Jackson integrals, (In preparation). · Zbl 1173.33013 · doi:10.1016/j.aim.2009.02.003
[7] K. Aomoto and Y. Kato, A \(q\)-analogue of de Rham cohomology associated with Jackson integrals, in Special functions ( Okayama , 1990), 30-62, Springer, Tokyo. · Zbl 0768.33019
[8] R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319. · Zbl 0572.33012 · doi:10.1090/memo/0319
[9] J.F. van Diejen, On certain multiple Bailey, Rogers and Dougall type summation formulas, Publ. Res. Inst. Math. Sci. 33 (1997), 483-508. · Zbl 0894.33007 · doi:10.2977/prims/1195145326
[10] R.A. Gustafson, Some \(q\)-beta and Mellin-Barnes integrals on compact Lie groups and Lie algebras, Trans. Amer. Math. Soc. 341 (1994), 69-119. · Zbl 0796.33012 · doi:10.2307/2154615
[11] M. Ito, Symmetry classification for Jackson integrals associated with irreducible reduced root systems, Compositio Math. 129 (2001), 325-340. · Zbl 0990.33016 · doi:10.1023/A:1012518910847
[12] M. Ito, Symmetry classification for Jackson integrals associated with the root system \(BC_ n\), Compositio Math. 136 (2003), 209-216. · Zbl 1094.33014 · doi:10.1023/A:1022892011301
[13] M. Ito, Askey-Wilson type integrals associated with root systems, Ramanujan J. (To appear). · Zbl 1114.33026 · doi:10.1007/s11139-006-9579-y
[14] M. Ito, \(q\)-difference shift for a \(BC_{n}\) type Jackson integral arising from ‘elementary’ symmetric polynomials, Adv. Math. (To appear). · Zbl 1098.33012 · doi:10.1016/j.aim.2005.06.008
[15] M. Ito, Another proof of Gustafson’s \(C_n\)-type summation formula via ‘elementary’ symmetric polynomials, Publ. Res. Inst. Math. Sci. (To appear). · Zbl 1210.33021 · doi:10.2977/prims/1166642114
[16] K.W.J. Kadell, A proof of the \(q\)-Macdonald-Morris conjecture for \(BC_{n}\), Mem. Amer. Math. Soc. 180 (1994), no. 516. · Zbl 0796.17003 · doi:10.1090/memo/0516
[17] C. Sabbah, Systèmes holonomes d’équations aux \(q\)-différences, in \(D\)-modules and microlocal geometry ( Lisbon , 1990), 125-147, de Gruyter, Berlin, 1993. · Zbl 0794.33012
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