Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Finite Aomoto-Ito-Macdonald sums

Author(s): Ana-Cecilia de la Maza
Journal: Proc. Amer. Math. Soc. 132 (2004), 2085-2094.
MSC (2000): Primary 33D67; Secondary 05A19, 11L03
Posted: February 6, 2004
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

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:

[AAR]
G. E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999. MR 2000g:33001

[A]
K. Aomoto, On elliptic product formulas for Jackson integrals associated with reduced root systems, J. Algebraic Combin. 8 (1998), 115-126. MR 2000b:33012

[B]
N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4-6, Hermann, Paris, 1968.MR 39:1590

[D]
J. F. van Diejen, On certain multiple Bailey, Rogers, and Dougall type summation formulas, Publ. Res. Inst. Math. Sci. 33 (1997), 483-508. MR 98j:33011

[DS]
J. F. van Diejen and V. P. Spiridonov, An elliptic Macdonald-Morris conjecture and multiple modular hypergeometric sums, Math. Res. Lett. 7 (2000), 729-746.MR 2002m:33022

[DV]
J. F. van Diejen and L. Vinet, The quantum dynamics of the compactified trigonometric Ruijsenaars-Schneider model, Commun. Math. Phys. 197 (1998), 33-74.MR 2000k:81123

[GR]
G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications 35, Cambridge Univ. Press, Cambridge, 1990. MR 91d:33034

[G1]
R. A. Gustafson, Multilateral summation theorems for ordinary and basic hypergeometric series in ${\rm U}(n)$, SIAM J. Math. Anal. 18 (1987), 1576-1596. MR 90e:33015

[G2]
-, 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), 185-224, Macmillan of India, New Delhi, 1989.MR 92k:33015

[I1]
M. Ito, A theta product formula for Jackson integrals associated with root systems, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), 60-61.MR 99b:33043

[I2]
-, On a theta product formula for Jackson integrals associated with root systems of rank two, J. Math. Anal. Appl. 216 (1997), 122-163.MR 99e:33023

[I3]
-, Symmetry classification for Jackson integrals associated with irreducible root systems, Compositio Math. 129 (2001), 325-340. MR 2002m:33021

[K]
J. Kaneko, $q$-Selberg integrals and Macdonald polynomials, Ann. Sci. École Norm. Sup. (4) 29 (1996), 583-637.MR 98k:33026

[M1]
I. G. Macdonald, Affine root systems and Dedekind's $\eta $-function, Invent. Math. 15 (1972), 91-143. MR 50:9996

[M2]
-, The Poincaré series of a Coxeter group, Math. Ann. 199 (1972), 161-174.MR 48:433

[M3]
-, Some conjectures for root systems, SIAM J. Math. Anal. 13 (1982), 988-1007.MR 84h:17006a

[M4]
-, A formal identity for affine root systems, preprint, 1996.

[M]
S. C. Milne, Basic hypergeometric series very well-poised in ${\rm U}(n)$, J. Math. Anal. Appl. 122 (1987), 223-256. MR 88h:33007

[ML]
S. C. Milne and G. M. Lilly, Consequences of the $A\sb l$ and $C\sb l$ Bailey transform and Bailey lemma, Formal Power Series and Algebraic Combinatorics (Montreal, PQ, 1992). Discrete Math. 139 (1995), 319-346.MR 96g:33025

[O]
E. M. Opdam, Some applications of hypergeometric shift operators, Invent. Math. 98 (1989), 1-18. MR 91h:33024

[R]
H. Rosengren, Karlsson-Minton type hypergeometric functions on the root system $C_n$, J. Math. Anal. Appl. 281 (2003), 332-345.

[S]
M. Schlosser, Multidimensional matrix inversions and multiple basic hypergeometric series associated to root systems, Special Functions and Differential Equations (Madras, 1997), 25-30, Allied Publ., New Delhi, 1998.MR 99i:33024

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 33D67, 05A19, 11L03

Retrieve articles in all Journals with MSC (2000): 33D67, 05A19, 11L03

Additional Information:

Ana-Cecilia de la Maza
Affiliation: Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile
Email: anace@inst-mat.utalca.cl

DOI: 10.1090/S0002-9939-04-07305-8
PII: S 0002-9939(04)07305-8
Keywords: Multivariate basic hypergeometric series, summation formulas, root systems
Received by editor(s): December 13, 2002
Received by editor(s) in revised form: April 3, 2003 and April 11, 2003
Posted: February 6, 2004
Additional Notes: This work was supported in part by the Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) Grant # 1010205 and the Programa formas cuadráticas of the Universidad de Talca.
Communicated by: John R. Stembridge
Copyright of article: Copyright 2004, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google