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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

∑k=−aa(−1)k(2ak+a)3=(3a)!(a!)3.{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

∑k∈Z(−1)k(a+ba+k)(b+cb+k)(c+ac+k)=(a+b+c)!a!b!c!{displaystyle sum _{kin 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

(b+cb−a)(c+ac−a)3F2(−2a,−a−b,−a−c;1+b−a,1+c−a;1){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 ₃F₂generalized hypergeometric series at 1, from (Dixon 1902):

3F2(a,b,c;1+a−b,1+a−c;1)=Γ(1+a/2)Γ(1+a/2−b−c)Γ(1+a−b)Γ(1+a−c)Γ(1+a)Γ(1+a−b−c)Γ(1+a/2−b)Γ(1+a/2−c).{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.

q-analogues

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

4ϕ3[a−qa1/2bc−a1/2aq/baq/c;q,qa1/2/bc]=(aq,aq/bc,qa1/2/b,qa1/2/c;q)∞(aq/b,aq/c,qa1/2,qa1/2/bc;q)∞{displaystyle ;_{4}phi _{3}left[{begin{matrix}a&-qa^{1/2}&b&c\&-a^{1/2}&aq/b&aq/cend{matrix}};q,qa^{1/2}/bcright]={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.

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, JFM 22.0258.01.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  


• 
  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 (27): 46–54, ISSN 0791-5578, MR 1185413, Zbl 0795.01009
  


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  Wilf, Herbert S. (1994), Generatingfunctionology (2nd ed.), Boston, MA: Academic Press, ISBN 0-12-751956-4, Zbl 0831.05001