A short proof of the Zeilberger-Bressoud 𝑞-Dyson theorem

Gessel, Ira; Xin, Guoce

doi:10.1090/S0002-9939-06-08224-4

A short proof of the Zeilberger-Bressoud $q$-Dyson theorem
HTML articles powered by AMS MathViewer

by Ira M. Gessel and Guoce Xin

Proc. Amer. Math. Soc. 134 (2006), 2179-2187

DOI: https://doi.org/10.1090/S0002-9939-06-08224-4

Published electronically: March 14, 2006

PDF | Request permission

Abstract:

We give a formal Laurent series proof of Andrews’s $q$-Dyson Conjecture, first proved by Zeilberger and Bressoud.

References

George E. Andrews, Problems and prospects for basic hypergeometric functions, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Math. Res. Center, Univ. Wisconsin, Publ. No. 35, Academic Press, New York, 1975, pp. 191–224. MR 0399528
George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
D. M. Bressoud and I. P. Goulden, Constant term identities extending the $q$-Dyson theorem, Trans. Amer. Math. Soc. 291 (1985), no. 1, 203–228. MR 797055, DOI 10.1090/S0002-9947-1985-0797055-8
Ivan Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Ann. of Math. (2) 141 (1995), no. 1, 191–216. MR 1314036, DOI 10.2307/2118632
I. J. Good, Short proof of a conjecture by Dyson, J. Mathematical Phys. 11 (1970), 1884. MR 258644, DOI 10.1063/1.1665339
J. Gunson, Proof of a conjecture by Dyson in the statistical theory of energy levels, J. Mathematical Phys. 3 (1962), 752–753. MR 148401, DOI 10.1063/1.1724277
Kevin W. J. Kadell, A proof of Andrews’ $q$-Dyson conjecture for $n=4$, Trans. Amer. Math. Soc. 290 (1985), no. 1, 127–144. MR 787958, DOI 10.1090/S0002-9947-1985-0787958-2
I. G. Macdonald, Some conjectures for root systems, SIAM J. Math. Anal. 13 (1982), no. 6, 988–1007. MR 674768, DOI 10.1137/0513070
Richard P. Stanley, The $q$-Dyson conjecture, generalized exponents, and the internal product of Schur functions, Combinatorics and algebra (Boulder, Colo., 1983) Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 81–94. MR 777696, DOI 10.1090/conm/034/777696
Richard P. Stanley, The stable behavior of some characters of $\textrm {SL}(n,\textbf {C})$, Linear and Multilinear Algebra 16 (1984), no. 1-4, 3–27. MR 768993, DOI 10.1080/03081088408817606
John R. Stembridge, A short proof of Macdonald’s conjecture for the root systems of type $A$, Proc. Amer. Math. Soc. 102 (1988), no. 4, 777–786. MR 934842, DOI 10.1090/S0002-9939-1988-0934842-5
Kenneth G. Wilson, Proof of a conjecture by Dyson, J. Mathematical Phys. 3 (1962), 1040–1043. MR 144627, DOI 10.1063/1.1724291
Guoce Xin, A residue theorem for Malcev-Neumann series, Adv. in Appl. Math. 35 (2005), no. 3, 271–293. MR 2164920, DOI 10.1016/j.aam.2005.02.004
Guoce Xin, A fast algorithm for MacMahon’s partition analysis, Electron. J. Combin. 11 (2004), no. 1, Research Paper 58, 20. MR 2097324, DOI 10.37236/1811
Doron Zeilberger, Proof of a conjecture of Chan, Robbins, and Yuen, Electron. Trans. Numer. Anal. 9 (1999), 147–148. Orthogonal polynomials: numerical and symbolic algorithms (Leganés, 1998). MR 1749805
Doron Zeilberger and David M. Bressoud, A proof of Andrews’ $q$-Dyson conjecture, Discrete Math. 54 (1985), no. 2, 201–224. MR 791661, DOI 10.1016/0012-365X(85)90081-0

AMS Website Down

Proceedings of the American Mathematical Society

A short proof of the Zeilberger-Bressoud $q$-Dyson theorem
HTML articles powered by AMS MathViewer

Abstract: