Finite Aomoto-Ito-Macdonald sums

de la Maza, Ana-Cecilia

doi:10.1090/S0002-9939-04-07305-8

Finite Aomoto-Ito-Macdonald sums
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Proc. Amer. Math. Soc. 132 (2004), 2085-2094 Request permission

Abstract:

We present finite truncations of the Aomoto-Ito-Macdonald sums associated with root systems through a two-step reduction procedure. The first reduction restricts the sum from the root lattice to a Weyl chamber; the second reduction arises after imposing a truncation condition on the parameters, and gives rise to a finite sum over a Weyl alcove.

References

George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
Kazuhiko Aomoto, On elliptic product formulas for Jackson integrals associated with reduced root systems, J. Algebraic Combin. 8 (1998), no. 2, 115–126. MR 1648460, DOI 10.1023/A:1008629309210
N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
Jan F. van Diejen, On certain multiple Bailey, Rogers and Dougall type summation formulas, Publ. Res. Inst. Math. Sci. 33 (1997), no. 3, 483–508. MR 1474700, DOI 10.2977/prims/1195145326
J. F. van Diejen and V. P. Spiridonov, An elliptic Macdonald-Morris conjecture and multiple modular hypergeometric sums, Math. Res. Lett. 7 (2000), no. 5-6, 729–746. MR 1809297, DOI 10.4310/MRL.2000.v7.n6.a6
J. F. van Diejen and L. Vinet, The quantum dynamics of the compactified trigonometric Ruijsenaars-Schneider model, Comm. Math. Phys. 197 (1998), no. 1, 33–74. MR 1646479, DOI 10.1007/s002200050442
George Gasper and Mizan Rahman, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR 1052153
R. A. Gustafson, Multilateral summation theorems for ordinary and basic hypergeometric series in $\textrm {U}(n)$, SIAM J. Math. Anal. 18 (1987), no. 6, 1576–1596. MR 911651, DOI 10.1137/0518114
R. A. Gustafson, The Macdonald identities for affine root systems of classical type and hypergeometric series very-well-poised on semisimple Lie algebras, Ramanujan International Symposium on Analysis (Pune, 1987) Macmillan of India, New Delhi, 1989, pp. 185–224. MR 1117471
Masahiko Ito, A theta product formula for Jackson integrals associated with root systems, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 4, 60–61. MR 1453532
Masahiko Ito, On a theta product formula for Jackson integrals associated with root systems of rank two, J. Math. Anal. Appl. 216 (1997), no. 1, 122–163. MR 1487257, DOI 10.1006/jmaa.1997.5665
Masahiko Ito, Symmetry classification for Jackson integrals associated with irreducible reduced root systems, Compositio Math. 129 (2001), no. 3, 325–340. MR 1868358, DOI 10.1023/A:1012518910847
Jyoichi Kaneko, $q$-Selberg integrals and Macdonald polynomials, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 5, 583–637. MR 1399617, DOI 10.24033/asens.1749
I. G. Macdonald, Affine root systems and Dedekind’s $\eta$-function, Invent. Math. 15 (1972), 91–143. MR 357528, DOI 10.1007/BF01418931
I. G. Macdonald, The Poincaré series of a Coxeter group, Math. Ann. 199 (1972), 161–174. MR 322069, DOI 10.1007/BF01431421
I. G. Macdonald, Some conjectures for root systems, SIAM J. Math. Anal. 13 (1982), no. 6, 988–1007. MR 674768, DOI 10.1137/0513070
—, A formal identity for affine root systems, preprint, 1996.
S. C. Milne, Basic hypergeometric series very well-poised in $\textrm {U}(n)$, J. Math. Anal. Appl. 122 (1987), no. 1, 223–256. MR 874970, DOI 10.1016/0022-247X(87)90355-6
Stephen C. Milne and Glenn M. Lilly, Consequences of the $A_l$ and $C_l$ Bailey transform and Bailey lemma, Discrete Math. 139 (1995), no. 1-3, 319–346. Formal power series and algebraic combinatorics (Montreal, PQ, 1992). MR 1336846, DOI 10.1016/0012-365X(94)00139-A
E. M. Opdam, Some applications of hypergeometric shift operators, Invent. Math. 98 (1989), no. 1, 1–18. MR 1010152, DOI 10.1007/BF01388841
H. Rosengren, Karlsson-Minton type hypergeometric functions on the root system $C_n$, J. Math. Anal. Appl. 281 (2003), 332–345.
Michael Schlosser, Multidimensional matrix inversions and multiple basic hypergeometric series associated to root systems, Special functions and differential equations (Madras, 1997) Allied Publ., New Delhi, 1998, pp. 25–30. MR 1659723