Disturbing the Dyson Conjecture (in a Generally GOOD Way)

Let  F n :=  F n  ( x 1 ,  x 2 , . . . ,  x n  ;  a 1 ,  a 2 , . . . ,  a n ) := ∏ 1≤i≠j≤n  (1-x i / x j )  aj  . In 1962, Freeman Dyson conjectured that the constant term in the expansion of  F n  is the multinomial coefficient ( a 1  +  a 2  + . . . +  a n )!/( a 1 !  a 2 ! . . .  a n !). In 1975, George Andrews extended Dyson's conjecture to a  q -analog. A particularly elegant proof of Dyson's conjecture was given by I. J. Good in 1970. Good's proof does not extend to the  q -analog, however, and the  q -Dyson conjecture was not settled until 1985 when Zeilberger and Bressoud proved it combinatorially.

Last March in the Experimental Mathematics Seminar, I demonstrated how a Maple package that I developed with Professor Zeilberger could be used to automatically conjecture and prove closed form expressions for coefficients in the expansion of  Fn besides the constant term, for  fixed n . The automated proofs are based on a generalization of Good's proof. In this lecture, I will discuss more recent work, where the "disturbed Dyson conjectures" and their proofs are extended to  symbolic n , and corresponding  q -analogs are conjectured.