The Chu-Vandermonde identity

(1)

(for ) is a special case of Gauss's hypergeometric theorem

			(2)
			(3)

(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)

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)

which is sometimes known as Vandermonde's convolution formula (Roman 1984). A special case gives the identity

(6)

The most famous special case follows from taking and using the identity in (6) to obtain

(7)

The identities

			(8)
			(9)
			(10)

are all special instances of the Chu-Vandermonde identity (Koepf 1998, p. 41).

Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, 1935.

Boros, G. and Moll, V. Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge, England: Cambridge University Press, 2004.

Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.

Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, pp. 130 and 181-182, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.

Roman, S. The Umbral Calculus. New York: Academic Press, p. 29, 1984.

CITE THIS AS:

Weisstein, Eric W. "Chu-Vandermonde Identity." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Chu-VandermondeIdentity.html