mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 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, which ...

. 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 ide ...

, 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

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s . 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, which ...

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 Atle Selberg. Selberg's integral formula When Re(\alpha) > 0, Re(\beta) > 0, Re(\gamma) > -\min \left(\frac 1n , \frac, \frac\right) ...

''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-analog, ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is ...

in terms of the

q-Pochhammer symbol In the mathematical field 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 ...

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