In mathematics, the Dyson conjecture is a conjecture about the constant term of certain Laurent polynomials, proved independently in 1962 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
\prod1\le(1-ti/t
ai | |
j) |
has constant term
(a1+a2+ … +an)! | |
a1!a2! … an! |
.
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
F(a1,...,an)=
nF(a | |
\sum | |
1,...,a |
i-1,...,an).
The case n = 3 of Dyson's conjecture follows from the Dixon identity.
and used a computer to find expressions for non-constant coefficients ofDyson's Laurent polynomial.
When all the values ai are equal to β/2, the constant term in Dyson's conjecture is the value of Dyson's integral
1 | |
(2\pi)n |
2\pi | |
\int | |
0 |
2\pi | |
… \int | |
0 |
\prod1\le
i\thetaj | |
|e |
i\thetak | |
-e |
|\betad\theta1 … d\thetan.
Dyson's integral is a special case of Selberg's integral after a change of variable and has value
\Gamma(1+\betan/2) | |
\Gamma(1+\beta/2)n |
which gives another proof of Dyson's conjecture in this special case.
found a q-analog of Dyson's conjecture, stating that the constant term of
\prod1\le\left(
xi | |
xj |
;q\right) | \left( | |
ai |
qxj | |
xi |
;q\right) | |
aj |
| ||||||||||
|
.
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 using doubly affine Hecke algebras.
Macdonald's form of Dyson's conjecture for root systems of type BC is closely related to Selberg's integral.