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 Amzi Clarence Dixon (July 6, 1854 – June 14, 1925) was a Baptist pastor, Bible expositor, and evangelist who was popular during the late 19th and the early 20th centuries. With R.A. Torrey, he edited an influential series of essays, publish ...

, some involving finite sums of products of three

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

s, and some evaluating a

hypergeometric sum In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, whic ...

. These identities famously follow from the

MacMahon Master theorem In mathematics, MacMahon's master theorem (MMT) is a result in enumerative combinatorics and linear algebra. It was discovered by Percy MacMahon and proved in his monograph ''Combinatory analysis'' (1916). It is often used to derive binomial iden ...

, and can now be routinely proved by computer algorithms .

Statements

The original identity, from , is :

\sum_^(-1)^^3 =\frac.

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

\sum_(-1)^k   = \frac

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

_3F_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 ₃''F''₂

generalized hypergeometric series In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, whic ...

at 1, from : :

\;_3F_2 (a,b,c;1+a-b,1+a-c;1)=
\frac
.

This holds for Re(1 + ''a'' − ''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 In mathematics, the Selberg integral is a generalization of Euler beta function to ''n'' dimensions introduced by . Selberg's integral formula When Re(\alpha) > 0, Re(\beta) > 0, Re(\gamma) > -\min \left(\frac 1n , \frac, \frac\right), we have : ...

''q''-analogues

A ''q''-analogue of Dixon's formula for the

basic hypergeometric series In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called ...

in terms of the

q-Pochhammer symbol In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer sym ...

is given by :

= \frac

where , ''qa''^1/2/''bc'', < 1.

References

* * * * * * {{citation , last=Wilf , first=Herbert S. , authorlink=Herbert Wilf , title=Generatingfunctionology , edition=2nd , location=Boston, MA , publisher=Academic Press , year=1994 , isbn=0-12-751956-4 , zbl=0831.05001 Enumerative combinatorics Factorial and binomial topics Hypergeometric functions Mathematical identities