The Chu-Vandermonde identity
 |
(1)
|
(for 
) is a special case of Gauss's
hypergeometric theorem
 |  |  |
(2)
|
 |  |  |
(3)
|
(which holds for ![Re[c-a-b]>0](/images/equations/Chu-VandermondeIdentity/Inline8.svg)
),
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).
