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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 [[Alfred Dixon|A. C. Dixon]], some involving finite sums of products of three [[binomial coefficient]]s, and some evaluating a [[hypergeometric sum]]. These identities famously follow from the [[MacMahon Master theorem]], and can now be routinely proved by computer algorithms {{harv|Ekhad|1990}}.
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| ==Statements==
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| The original identity, from {{harv|Dixon|1891}}, is
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| :<math>\sum_{k=-a}^{a}(-1)^{k}{2a\choose k+a}^3 =\frac{(3a)!}{(a!)^3}.</math>
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| A generalization, also sometimes called Dixon's identity, is
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| :<math>\sum_{k=-a}^a(-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>
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| where ''a'', ''b'', and ''c'' are non-negative integers {{harv|Wilf|1994|p=156}}.
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| The sum on the left can be written as the terminating well-poised hypergeometric series
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| :<math>{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>
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| and the identity follows as a limiting case (as ''a'' tends to an integer) of
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| Dixon's theorem evaluating a well-poised <sub>3</sub>''F''<sub>2</sub> [[generalized hypergeometric series]] at 1, from {{harv|Dixon|1902}}:
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| :<math>\;_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)}
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| {\Gamma(1+a)\Gamma(1+a-b-c)\Gamma(1+a/2-b)\Gamma(1+a/2-c)}.</math>
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| This holds for Re(1 + {{frac|1|2}}''a'' − ''b'' − ''c'') > 0. As ''c'' tends to −∞ it reduces to [[Kummer's formula]] for the hypergeometric function <sub>2</sub>F<sub>1</sub> at −1. Dixon's theorem can be deduced from the evaluation of the [[Selberg integral]].
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| ==q-analogues==
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| A ''q''-analogue of Dixon's formula for the [[basic hypergeometric series]] in terms of the [[q-Pochhammer symbol]] is given by
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| :<math>\;_{4}\phi_3 \left[\begin{matrix} | |
| a & -qa^{1/2} & b & c \\
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| &-a^{1/2} & aq/b & aq/c \end{matrix}
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| ; q,qa^{1/2}/bc \right] =
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| \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}
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| </math>
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| where |''qa''<sup>1/2</sup>/''bc''| < 1.
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| ==References==
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| {{reflist}}
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| *{{citation |first=A.C. |last=Dixon | authorlink=Alfred Cardew Dixon | title= On the sum of the cubes of the coefficients in a certain expansion by the binomial theorem |journal= [[Messenger of Mathematics]] |volume=20 | pages= 79–80 |year=1891 | jfm=22.0258.01 }}
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| *{{citation |first=A.C. |last=Dixon | authorlink=Alfred Cardew Dixon |title= Summation of a certain series |journal= Proc. London Math. Soc. |volume=35 |issue=1 |pages= 284–291 |year=1902 | doi=10.1112/plms/s1-35.1.284 | jfm=34.0490.02 }}
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| *{{Citation | last1=Ekhad | first1=Shalosh B. | authorlink=Shalosh Ekhad | title=A very short proof of Dixon's theorem | doi=10.1016/0097-3165(90)90014-N | mr=1051787 | zbl=0707.05007 | year=1990 | journal=Journal of Combinatorial Theory. Series A | issn=1096-0899 | volume=54 | issue=1 | pages=141–142}}
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| *{{Citation | last1=Gessel | first1=Ira | last2=Stanton | first2=Dennis | title=Short proofs of Saalschütz's and Dixon's theorems | doi=10.1016/0097-3165(85)90026-3 | mr=773560 | zbl=0559.05008 | year=1985 | journal=Journal of Combinatorial Theory. Series A | issn=1096-0899 | volume=38 | issue=1 | pages=87–90}}
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| *{{Citation | last1=Ward | first1=James | title=100 years of Dixon's identity | mr=1185413 | zbl=0795.01009 | year=1991 | journal=Irish Mathematical Society Bulletin | issn=0791-5578 | issue=27 | pages=46–54}}
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| * {{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 }}
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| [[Category:Enumerative combinatorics]]
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| [[Category:Factorial and binomial topics]]
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| [[Category:Hypergeometric functions]]
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| [[Category:Mathematical identities]]
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Hi there! :) My name is Kasey, I'm a student studying Athletics and Physical Education from Castres, France.
Feel free to surf to my web blog: woman cancer