Zeilberger-Bressoud Theorem

Dyson (1962abc) conjectured that the constant term in the Laurent series

(1)

is the multinomial coefficient

(2)

based on a problem in particle physics. The theorem is called Dyson's conjecture, and was proved by Wilson (1962) and independently by Gunson (1962). A definitive proof was subsequently published by Good (1970).

A -analog of this theorem (Andrews 1975) states that the coefficient of  in

(3)

where

(4)

is given by

(5)

This can also be stated in the form that the constant term of

(6)

is the q-multinomial coefficient

(7)

where  is the q-factorial. The amazing proof of this theorem was given by Zeilberger and Bressoud (1985).

The full theorem reduces to Dyson's version when . It also gives the q-analog of Dixon's theorem as

(8)

(Andrews 1975, 1986), where  is a q-binomial coefficient. With  and , it gives the beautiful and well-known identity

(9)

(Andrews 1986).

REFERENCES:

Andrews, G. E. "Problems and Prospects for Basic Hypergeometric Functions." In The Theory and Application of Special Functions (Ed. R. Askey). New York: Academic Press, pp. 191-224, 1975.

Andrews, G. E. "The Zeilberger-Bressoud Theorem." §4.3 in q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., pp. 36-38, 1986.

Dyson, F. "Statistical Theory of the Energy Levels of Complex Systems. I." J. Math. Phys. 3, 140-156, 1962a.

Dyson, F. "Statistical Theory of the Energy Levels of Complex Systems. II." J. Math. Phys. 3, 157-165, 1962b.

Dyson, F. "Statistical Theory of the Energy Levels of Complex Systems. III." J. Math. Phys. 3, 166-175, 1962c.

Good, I. J. "Short Proof of a Conjecture by Dyson." J. Math. Phys. 11, 1884, 1970.

Gunson, J. "Proof of a Conjecture of Dyson in the Statistical Theory of Energy Levels." J. Math. Phys. 3, 752-753, 1962.

Wilson, K. G. "Proof of a Conjecture by Dyson." J. Math. Phys. 3, 1040-1043, 1962.

Zeilberger, D. and Bressoud, D. M. "A Proof of Andrews' -Dyson Conjecture." Disc. Math. 54, 201-224, 1985.