The Chu-Vandermonde identity
 |
(1) |
(for 
)
is a special case of Gauss's hypergeometric
theorem
(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
are all special instances of the Chu-Vandermonde identity (Koepf 1998, p. 41).
See also
Binomial Theorem, Gauss's Hypergeometric Theorem, q-Chu-Vandermonde Identity, Umbral Calculus
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References
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.
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Cite this as:
Weisstein, Eric W. "Chu-Vandermonde Identity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Chu-VandermondeIdentity.html