A Dyson constant term orthogonality relation
For \(n\geq 2\) and \(a_1,a_2,\dots, a_n\geq 0\), set \[ f_n(a_1,\dots, a_n; t_1,\dots, t_n)= \prod_{1\leq i<j\leq n}(1- t_i/t_j)^{a_i}(1- t_j/t_i)^{a_j}. \] Let \(r\geq 1\). Denote the compositions of \(r\) by \((w_1,\dots, w_n)\sim r\Leftrightarrow r= w_1+\cdots+ w_n\), \(w_1,\dots, w_n\geq 0\). Denote the complete homogeneous function of order \(r\) by \[ h_r(s_1,\dots, s_n)= \sum_{(w_1,\dots, w_n)\sim r} \prod^n_{i=1} s^{w_i}_i. \] Let \((\widetilde t^{a_1}_1,\dots, \widetilde t^{a_n}_n)\) denote the set of variables which is obtained by repeating variable \(t_i\) exactly \(a_i\) times for \(1\leq i\leq n\). Set \[ F^{-w_1,\dots, -w_n}_{n,r}(a_1,\dots, a_n)= \prod^n_{i= 1} t^{-w_i}_i h_r(\widetilde t^{a_1}_1,\dots, \widetilde t^{a_n}_n) f_n(a_1,\dots, a_n; t_1,\dots, t_n). \] The paper proves the following: if \(n\geq 2\), \(a_1,a_2,\dots, a_n\geq 0\), and \((w_1,\dots, w_n)\sim r\), then \[ \text{if }w_1,\dots, w_n< r,\text{ then } F^{-w_1,\dots, -w_n}_{n,r}(a_1,\dots, a_n)= 0,\tag{1} \] \[ F^{-r,0,\dots,0}_{n,r}(a_1,\dots, a_n)= {a_1\over (1+ a_2+\cdots+ a_n)_r} {(a_1+\cdots+ a_n+ r- 1)!\over a_1!\cdots a_n!},\tag{2} \] (3) conjectures a \(q\)-analogue of (1) and (2) which has been verified for \(n=3\), \(0\leq a_1,a_2,a_3\leq 4\) by computer. This is closely related to Dyson's constant term identity and its \(q\)-analogue conjectured by \textit{G. E. Andrews} and proved by \textit{D. Zeilberger} and \textit{D. M. Bressoud} [Discrete Math. 54, 201-224 (1985; Zbl 0565.33001)]. The proof is based on the identity \(1= \sum j= 1^n\prod^n_{i=1, i\neq j}(1- t_j/t_i)^{-1}\) like I. J. Good's proof to Dyson's conjecture; see [Short proof of a conjecture of Dyson, J. Math. Phys. 11, 1884 (1970)].
scientific article; zbMATH DE number 1422987