Dyson conjecture

Remarks

Freeman Dyson stated this as a conjecture in 1962, in work on the statistical theory of energy levels. It was proved the same year by Gunson and by Wilson, and Good gave a short proof in 1970. Good's proof shows that both sides satisfy the recurrence $F(a) = \sum_{i} F(a - e_i)$ when all $a_i \ge 1$, and that both reduce to the case of $n - 1$ variables when some $a_i = 0$. The recurrence for the left side comes from the identity $\sum_{i=1}^n \prod_{j \ne i} \left(1 - \frac{x_j}{x_i}\right)^{-1} = 1$, a case of Lagrange interpolation. Example: $n = 2$. The constant term of $(1 - x_1/x_2)^{a_1} (1 - x_2/x_1)^{a_2}$ collects the products of the terms $(x_1/x_2)^k$ and $(x_2/x_1)^k$, and equals $\sum_{k} \binom{a_1}{k}\binom{a_2}{k} = \binom{a_1 + a_2}{a_1}$ by Vandermonde's identity.

Going deeper

For a student

The case where all $a_i$ equal $k$ gives $\frac{(nk)!}{(k!)^n}$, which computes the normalizing constant of the circular ensembles of random matrices through an integral over the torus. Andrews proposed a $q$-analogue, proved by Zeilberger and Bressoud. The equal-parameter case is the type $A$ case of the Macdonald constant term conjecture.