Theorem about the constant term of certain Laurent polynomials
In mathematics, the Dyson conjecture (
Freeman Dyson1962) 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 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.
Sills & Zeilberger (2006) and (
Sills 2006) 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
Andrews (1975) found a
q-analog 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
Zeilberger & Bressoud (1985), using a combinatorial approach inspired by
previous work of
Ira Gessel and
Dominique Foata. A shorter proof, using formal Laurent series, was given in 2004 by Ira Gessel and Guoce Xin, and
an even shorter proof, using a quantitative form, due to Karasev and Petrov, and independently to Lason, of Noga Alon's Combinatorial Nullstellensatz,
was given in 2012 by Gyula Karolyi and Zoltan Lorant Nagy.
The latter method was extended, in 2013, by Shalosh B. Ekhad and Doron Zeilberger to derive explicit expressions of any specific coefficient, not just the
constant term, see
http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/qdyson.html, for detailed references.
Macdonald conjectures
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.
Macdonald's form of Dyson's conjecture for root systems of type BC is closely related to
Selberg's integral.
References
Andrews, George E. (1975), "Problems and prospects for basic hypergeometric functions", Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), Boston, MA:
Academic Press, pp. 191–224,
MR0399528
Cherednik, I. (1995), "Double Affine Hecke Algebras and Macdonald's Conjectures", The Annals of Mathematics, 141 (1): 191–216,
doi:
10.2307/2118632,
JSTOR2118632