Dixon's identity

Short description: On finite sums of products of three binomial coefficients, and a hypergeometric sum

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).

Statements

The original identity, from (Dixon 1891), is

[math]\displaystyle{ \sum_{k=-a}^{a}(-1)^{k}{2a\choose k+a}^3 =\frac{(3a)!}{(a!)^3}. }[/math]

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

[math]\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!} }[/math]

where a, b, and c are non-negative integers (Wilf 1994). The sum on the left can be written as the terminating well-poised hypergeometric series

[math]\displaystyle{ {b+c\choose b-a}{c+a\choose c-a}{}_3F_2(-2a,-a-b,-a-c;1+b-a,1+c-a;1) }[/math]

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):

[math]\displaystyle{ \;_3F_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)}. }[/math]

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.

q-analogues

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

[math]\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} }[/math]

where |qa^1/2/bc| < 1.

References

• 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 
• 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, https://zenodo.org/record/1433433 
• 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 
• 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 
• 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 
• Wilf, Herbert S. (1994), Generatingfunctionology (2nd ed.), Boston, MA: Academic Press, ISBN 0-12-751956-4