In mathematics, the

**Dyson conjecture** 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.

## Dyson conjecture

The Dyson conjecture states that the

Laurent polynomial
has constant term

The conjecture was first proved independently by and . 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.

and used a computer to find expressions for non-constant coefficients of

Dyson's Laurent polynomial.

## Dyson integral

When all the values

*a*_{i} 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.

*q*-Dyson conjecture

found a

q-analogRoughly speaking, in mathematics, specifically in the areas of combinatorics and special functions, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1...

of Dyson's conjecture, stating that the constant term of

is

Here (

*a*;

*q*)

_{n} is the

q-Pochhammer symbol.

This conjecture reduces to Dyson's conjecture for

*q*=1, and was proved by .

## Macdonald conjectures

extended the conjecture to arbitrary finite or affine

root systemIn mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras...

s, with Dyson's original conjecture corresponding to

the case of the

*A*_{n−1} root system and Andrews's conjecture corresponding to the affine

*A*_{n−1} root system. Macdonald reformulated these conjectures as conjectures about the norms of

Macdonald polynomialIn mathematics, Macdonald polynomials Pλ are a family of orthogonal polynomials in several variables, introduced by...

s. Macdonald's conjectures were proved by using doubly affine Hecke algebras.

MacdonaldIan Grant Macdonald is a British mathematician known for his contributions to symmetric functions, special functions, Lie algebra theory and other aspects of algebraic combinatorics ....

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