Autour de la conjecture de Dyson. (Around the conjecture of Dyson) (Q1825871)

Given an n-tuple of positive integers, \((a_ 1,...,a_ n)\), let M be the set of words in \(a_ 1\) \(1's,...,a_ n\) n's and for \(w\in M\), let \(a_{ij}(w)\) be the number of i's to the left of the first j in w. Let \[ \phi_ w=\prod_{1\leq i\neq j\leq n}(1-\frac{x_ i}{x_ j})^{a_{ij}(w)},\quad \Phi =\prod_{1\leq i\neq j\leq n}(1-\frac{x_ i}{x_ j})^{a_ i}. \] The author proves that \(\Phi =\sum \phi_ w\), \(w\in M\), from which it is simple to derive Dyson's constant term conjecture [see \textit{I. G. Macdonald}, SIAM J. Math. Anal. 13, 988-1007 (1982; Zbl 0498.17006)]. The problems that arise in seeking a q-analog of this are discussed.