In mathematics, Dixon's identity (or Dixon's theorem or Dixon's formula) is any of several different but closely related identities proved by A. C. Dixon, some involving finite sums of products of three binomial coefficients, and some evaluating a hypergeometric sum. These identities famously follow from the MacMahon Master theorem, and can now be routinely proved by computer algorithms (Ekhad 1990).
The original identity, from (Dixon 1891), is
A generalization, also sometimes called Dixon's identity, is
where a, b, and c are non-negative integers (Wilf 1994, p. 156). The sum on the left can be written as the terminating well-poised hypergeometric series
This holds for Re(1 + 1⁄2a − b − c) > 0. As c tends to −∞ it reduces to Kummer's formula for the hypergeometric function 2F1 at −1. Dixon's theorem can be deduced from the evaluation of the Selberg integral.
where |qa1/2/bc| < 1.
- Dixon, A.C. (1891), "On the sum of the cubes of the coefficients in a certain expansion by the binomial theorem", Messenger of Mathematics 20: 79–80, JFM 22.0258.01
- Dixon, A.C. (1902), "Summation of a certain series", Proc. London Math. Soc. 35 (1): 284–291, doi:10.1112/plms/s1-35.1.284, JFM 34.0490.02
- Ekhad, Shalosh B. (1990), "A very short proof of Dixon's theorem", Journal of Combinatorial Theory. Series A 54 (1): 141–142, doi:10.1016/0097-3165(90)90014-N, ISSN 1096-0899, MR 1051787, Zbl 0707.05007
- Gessel, Ira; Stanton, Dennis (1985), "Short proofs of Saalschütz's and Dixon's theorems", Journal of Combinatorial Theory. Series A 38 (1): 87–90, doi:10.1016/0097-3165(85)90026-3, ISSN 1096-0899, MR 773560, Zbl 0559.05008
- Ward, James (1991), "100 years of Dixon's identity", Irish Mathematical Society Bulletin (27): 46–54, ISSN 0791-5578, MR 1185413, Zbl 0795.01009
- Wilf, Herbert S. (1994), Generatingfunctionology (2nd ed.), Boston, MA: Academic Press, ISBN 0-12-751956-4, Zbl 0831.05001