In mathematics, the Dyson conjecture (Freeman Dyson 1962) is a conjecture about the constant term of certain Laurent polynomials, proved by Wilson and Gunson. Andrews generalized it to the q-Dyson conjecture, proved by Zeilberger and Bressoud and sometimes called the Zeilberger–Bressoud theorem. Macdonald generalized it further to more general root systems with the Macdonald constant term conjecture, proved by Cherednik.
The Dyson conjecture states that the Laurent polynomial
has constant term
The conjecture was first proved independently by Wilson (1962) and Gunson (1962). 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.
When all the values ai are equal to β/2, the constant term in Dyson's conjecture is the value of Dyson's integral
Dyson's integral is a special case of Selberg's integral after a change of variable and has value
which gives another proof of Dyson's conjecture in this special case.
Macdonald (1982) extended the conjecture to arbitrary finite or affine root systems, with Dyson's original conjecture corresponding to the case of the An−1 root system and Andrews's conjecture corresponding to the affine An−1 root system. Macdonald reformulated these conjectures as conjectures about the norms of Macdonald polynomials. Macdonald's conjectures were proved by (Cherednik 1995) using doubly affine Hecke algebras.
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