Zeilberger-Bressoud Theorem

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

(1)

(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

$epsilon_(ij)={1 for i<j; q for i>j$

(4)

is given by

(5)

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

product_(1<=i<j<=n)(1-x_i/x_j)(1-qx_i/q_j)...(1-q^(a_i-1)x_i/x_j) ×(1-qx_j/x_i)(1-q^2x_j/x_i)...(1-q^(a_j)x_j/x_i),

(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

sum_(k=-infty)^infty(-1)^kq^(k(3k+1)/2)[b+c; c+k]_q[c+a; a+k]_q[a+b; b+k]_q =((q;q)_(a+b+c))/((q;q)_a(q;q)_b(q;q)_c)

(8)

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

(9)

(Andrews 1986).

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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'