Dyson conjecture

Dyson conjecture

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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 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.

q-Dyson conjecture


found a q-analog
Q-analog
Roughly 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 system
Root system
In 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 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 polynomial
Macdonald polynomial
In 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.

Macdonald
Ian G. Macdonald
Ian 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.