Dyson conjecture

Good (1970) later found a short proof, by observing that the Laurent polynomials, and therefore their constant terms, satisfy the recursion relations The case n = 3 of Dyson's conjecture follows from the Dixon identity.

Andrews (1975) found a q-analog of Dyson's conjecture, stating that the constant term of is Here (a;q)n is the q-Pochhammer symbol.

A shorter proof, using formal Laurent series, was given in 2004 by Ira Gessel and Guoce Xin, and an even shorter proof, using a quantitative form, due to Karasev and Petrov, and independently to Lason, of Noga Alon's Combinatorial Nullstellensatz, was given in 2012 by Gyula Karolyi and Zoltan Lorant Nagy.

The latter method was extended, in 2013, by Shalosh B. Ekhad and Doron Zeilberger to derive explicit expressions of any specific coefficient, not just the constant term; see http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/qdyson.html, for detailed references.

Macdonald's form of Dyson's conjecture for root systems of type BC is closely related to Selberg's integral.