Disturbing the Dyson conjecture, in a generally GOOD way
| Autor: | Andrew V. Sills |
|---|---|
| Rok vydání: | 2006 |
| Předmět: |
Zeilberger–Bressoud theorem
Conjecture Extension (predicate logic) Theoretical Computer Science Combinatorics Dyson conjecture Computational Theory and Mathematics Special functions Product (mathematics) q-Dixon sum FOS: Mathematics Mathematics - Combinatorics Multinomial theorem Discrete Mathematics and Combinatorics q-Dyson conjecture Combinatorics (math.CO) Hypergeometric function Statistical theory Mathematics |
| Zdroj: | Journal of Combinatorial Theory, Series A. 113(7):1368-1380 |
| ISSN: | 0097-3165 |
| DOI: | 10.1016/j.jcta.2005.12.005 |
| Popis: | Dyson's celebrated constant term conjecture ({\em J. Math. Phys.}, 3 (1962): 140--156) states that the constant term in the expansion of $\prod_{1\leqq i\neq j\leqq n} (1-x_i/x_j)^{a_j}$ is the multinomial coefficient $(a_1 + a_2 + \cdots + a_n)!/ (a_1! a_2! \cdots a_n!)$. The definitive proof was given by I. J. Good ({\em J. Math. Phys.}, 11 (1970) 1884). Later, Andrews extended Dyson's conjecture to a $q$-analog ({\em The Theory and Application of Special Functions}, (R. Askey, ed.), New York: Academic Press, 191--224, 1975.) In this paper, closed form expressions are given for the coefficients of several other terms in the Dyson product, and are proved using an extension of Good's idea. Also, conjectures for the corresponding $q$-analogs are supplied. Finally, perturbed versions of the $q$-Dixon summation formula are presented. 16 pages |
| Databáze: | OpenAIRE |
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