The Chu-Vandermonde identity
(1)
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(for ) is a special case of Gauss's hypergeometric theorem
(2)
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(3)
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(which holds for ), with equal to a negative integer . Here, is a hypergeometric function, is the Pochhammer symbol, and is a gamma function (Bailey 1935, p. 3; Koepf 1998, p. 32). The identity is sometimes also called Vandermonde's theorem.
The identity
(4)
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for an integer, where is a binomial coefficient and is again the Pochhammer symbol, is sometimes also known as the Chu-Vandermonde identity (Koepf 1998, p. 42), or sometimes Vandermonde's formula (Boros and Moll 2004, p. 18). Equation (4) can be written as
(5)
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which is sometimes known as Vandermonde's convolution formula (Roman 1984). A special case gives the identity
(6)
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The most famous special case follows from taking and using the identity in (6) to obtain
(7)
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The identities
(8)
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(9)
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(10)
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are all special instances of the Chu-Vandermonde identity (Koepf 1998, p. 41).