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Dyson's Conjecture

Based on a problem in particle physics, Dyson (1962abc) conjectured that the constant term in the Laurent series

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

is the multinomial coefficient

((a_1+a_2+...+a_n)!)/(a_1!a_2!...a_n!).

The theorem was proved by Wilson (1962) and independently by Gunson (1962). A definitive proof was subsequently published by Good (1970).

See also

Macdonald's Constant-Term Conjecture, Zeilberger-Bressoud Theorem

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References

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.

Referenced on Wolfram|Alpha

Dyson's Conjecture

Cite this as:

Weisstein, Eric W. "Dyson's Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DysonsConjecture.html

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