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Zeilberger-Bressoud Theorem

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

product_(1<=i!=j<=n)(1-(x_i)/(x_j))^(a_i)
(1)

is the multinomial coefficient

((a_1+a_2+...+a_n)!)/(a_1!a_2!...a_n!),
(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 q-analog of this theorem (Andrews 1975) states that the coefficient of x_1^0x_2^0...x_n^0 in

product_(1<=i!=j<=n)((x_i)/(x_j)epsilon_(ij);q)_(a_i)
(3)

where

epsilon_(ij)={1 for i<j; q for i>j
(4)

is given by

((q;q)_(a_1+a_2+...+a_n))/((q;q)_(a_1)(q;q)_(a_2)...(q;q)_(a_n)).
(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

([a_1+...+a_n]!)/([a_1]!...[a_n]!),
(7)

where [n]! 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 q=1. 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 [n; k]_q is a q-binomial coefficient. With q=1 and a=b=c=p, it gives the beautiful and well-known identity

sum_(k=0)^(2p)(-1)^k(2p; k)^3=((-1)^p(3p)!)/((p!)^3)
(9)

(Andrews 1986).

See also

Dixon's Theorem, q-Multinomial Coefficient, Macdonald's Constant-Term Conjecture, Multinomial Coefficient

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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' q-Dyson Conjecture." Disc. Math. 54, 201-224, 1985.

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Zeilberger-Bressoud Theorem

Cite this as:

Weisstein, Eric W. "Zeilberger-Bressoud Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Zeilberger-BressoudTheorem.html

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