Macdonald's constant term conjectures are related to root systems of Lie algebras (Macdonald 1982, Andrews
1986). They can be regarded as generalizations of Dyson's
conjecture (Dyson 1962), its -analog due to Andrews, and Mehta's conjecture (Mehta 2004).
The simplest of these states that if is a root system, then the
constant term in , where is a nonnegative integer,
is ,
where the are fixed integer parameters of the root
system corresponding to the fundamental invariants of the Weyl
group of (Andrews 1986, p. 41).

Opdam (1989) proved the case for all root systems. The general conjecture had remained
"almost proved" for some time, since the infinite families were accomplished
by Zeilberger-Bressoud (), Kadell (, ), and Gustafson (, ), while the exceptional cases were done by Zeilberger and
(independently) Habsieger (), Zeilberger ( dual), and Garvan and Gonnet ( and dual), using Zeilberger's method. This left only the three
root systems (, , ) which were infeasible to address using existing computers.
In the meanwhile, however, Cherednik (1993) proved the constant term conjectures
for all root systems using a methodology not dependent on classification.

A special case of the constant-term conjecture is given by the assertion that the constant term in

(1)

is .
Another special case asserts that the constant term in

Andrews, G. E. "The Macdonald Conjectures." §4.5 in q-Series:
Their Development and Application in Analysis, Number Theory, Combinatorics, Physics,
and Computer Algebra. Providence, RI: Amer. Math. Soc., pp. 40-42, 1986.Cherednik,
I. "The Macdonald Constant-Term Conjecture." Duke Math. J.70,
165-177, 1993.Cherednik, I. "The Macdonald Constant-Term Conjecture."
Internat. Math. Res. Not., No. 6, 165-177, 1993.Dyson, F.
"Statistical Theory of the Energy Levels of Complex Systems. I." J.
Math. Phys.3, 140-156, 1962.Macdonald, I. G. "Affine
Root Systems and Dedekind's -Function." Invent. Math.15, 91-143,
1972.Macdonald, I. G. "Some Conjectures for Root Systems."
SIAM J. Math. Anal.13, 988-1007, 1982.Mehta, M. L.
Random
Matrices, 3rd ed. New York: Academic Press, 2004.Opdam, E. M.
"Some Applications of Hypergeometric Shift Operators." Invent. Math.98,
1-18, 1989.Stanton, D. "Sign Variations of the Macdonald Identities."
SIAM J. Math. Anal.17, 1454-1460, 1986.