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Chu-Vandermonde Identity


The Chu-Vandermonde identity

 _2F_1(-n,b;c;1)=((c-b)_n)/((c)_n)
(1)

(for n in Z^+) is a special case of Gauss's hypergeometric theorem

_2F_1(a,b;c;1)=((c-b)_(-a))/((c)_(-a))
(2)
=(Gamma(c)Gamma(c-a-b))/(Gamma(c-a)Gamma(c-b))
(3)

(which holds for R[c-a-b]>0), with a equal to a negative integer -n. Here, _2F_1(a,b;c;z) is a hypergeometric function, (a)_n=a(a+1)...(a+n-1) is the Pochhammer symbol, and Gamma(z) is a gamma function (Bailey 1935, p. 3; Koepf 1998, p. 32). The identity is sometimes also called Vandermonde's theorem.

The identity

 (x+a)_n=sum_(k=0)^n(n; k)(x)_k(a)_(n-k)
(4)

for n an integer, where (n; k) is a binomial coefficient and (a) 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

 (x+a; n)=sum_(k=0)^n(x; k)(a; n-k),
(5)

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

 sum_(l=0)^(max(k,n))(m; k-l)(n; l)=(m+n; k).
(6)

The most famous special case follows from taking m=k=n and using the identity (n; n-l)=(n; l) in (6) to obtain

 sum_(l=0)^n(n; l)^2=(2n; n).
(7)

The identities

sum_(k=0)^(n)(a; k)(b; n-k)=(a+b; n)
(8)
sum_(k=0)^(n)(n; k)(s; t-k)=(n+s; t)
(9)
sum_(k=0)^(n)(n; k)(s; t+k)=(n+s; n+t)
(10)

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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Chu-Vandermonde Identity

Cite this as:

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

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