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 .
The original identity, from, is
a | |
\sum | |
k=-a |
(-1)k{2a\choosek+a}3=
(3a)! | |
(a!)3 |
.
A generalization, also sometimes called Dixon's identity, is
\sumk\inZ(-1)k{a+b\choosea+k}{b+c\chooseb+k}{c+a\choosec+k}=
(a+b+c)! | |
a!b!c! |
where a, b, and c are non-negative integers . The sum on the left can be written as the terminating well-poised hypergeometric series
{b+c\chooseb-a}{c+a\choosec-a}{}3F2(-2a,-a-b,-a-c;1+b-a,1+c-a;1)
3F2(a,b,c;1+a-b,1+a-c;1)=
\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) |
.
A q-analogue of Dixon's formula for the basic hypergeometric series in terms of the q-Pochhammer symbol is given by
4\varphi3\left[\begin{matrix}a&-qa1/2&b&c\ &-a1/2&aq/b&aq/c\end{matrix};q,qa1/2/bc\right]=
(aq,aq/bc,qa1/2/b,qa1/2/c;q)infty | |
(aq/b,aq/c,qa1/2,qa1/2/bc;q)infty |