Dixon's identity

The original identity, from (Dixon 1891), is

${\displaystyle \sum _{k=-a}^{a}(-1)^{k}{2a \choose k+a}^{3}={\frac {(3a)!}{(a!)^{3}}}.}$

A generalization, also sometimes called Dixon's identity, is

${\displaystyle \sum _{k\in \mathbb {Z} }(-1)^{k}{a+b \choose a+k}{b+c \choose b+k}{c+a \choose c+k}={\frac {(a+b+c)!}{a!b!c!}}}$

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

${\displaystyle {b+c \choose b-a}{c+a \choose c-a}{}_{3}F_{2}(-2a,-a-b,-a-c;1+b-a,1+c-a;1)}$

and the identity follows as a limiting case (as a tends to an integer) of Dixon's theorem evaluating a well-poised ₃F₂ generalized hypergeometric series at 1, from (Dixon 1902):

${\displaystyle \;_{3}F_{2}(a,b,c;1+a-b,1+a-c;1)={\frac {\Gamma (1+a/2)\Gamma (1+a/2-b-c)\Gamma (1+a-b)\Gamma (1+a-c)}{\Gamma (1+a)\Gamma (1+a-b-c)\Gamma (1+a/2-b)\Gamma (1+a/2-c)}}.}$

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.

A q-analogue of Dixon's formula for the basic hypergeometric series in terms of the q-Pochhammer symbol is given by

${\displaystyle \;_{4}\varphi _{3}\left[{\begin{matrix}a&-qa^{1/2}&b&c\\&-a^{1/2}&aq/b&aq/c\end{matrix}};q,qa^{1/2}/bc\right]={\frac {(aq,aq/bc,qa^{1/2}/b,qa^{1/2}/c;q)_{\infty }}{(aq/b,aq/c,qa^{1/2},qa^{1/2}/bc;q)_{\infty }}}}$

where |qa^1/2/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 0773560, Zbl 0559.05008
• Ward, James (1991), "100 years of Dixon's identity", Irish Mathematical Society Bulletin, 0027 (27): 46–54, doi:10.33232/BIMS.0027.46.54, 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