 Vandermonde's identity

For the expression for a special determinant, see Vandermonde matrix.
In combinatorics, Vandermonde's identity, or Vandermonde's convolution, named after AlexandreThéophile Vandermonde (1772), states that
for binomial coefficients. This identity was given already in 1303 by the Chinese mathematician Zhu Shijie (Chu ShiChieh). See Askey 1975, pp. 59–60 for the history.
There is a qanalog to this theorem called the qVandermonde identity.
Contents
Algebraic proof
In general, the product of two polynomials with degrees m and n, respectively, is given by
where we use the convention that a_{i} = 0 for all integers i > m and b_{j} = 0 for all integers j > n. By the binomial theorem,
Using the binomial theorem also for the exponents m and n, and then the above formula for the product of polynomials, we obtain
where the above convention for the coefficients of the polynomials agrees with the definition of the binomial coefficients, because both give zero for all i > m and j > n, respectively.
By comparing coefficients of x^{r}, Vandermonde's identity follows for all integers r with 0 ≤ r ≤ m + n. For larger integers r, both sides of Vandermonde's identity are zero due to the definition of binomial coefficients.
Combinatorial proof
Vandermonde's identity also admits a more combinatoricsflavored double counting proof, as follows. Suppose a committee in the US Senate consists of m Democrats and n Republicans. In how many ways can a subcommittee of r members be formed? The answer is of course
But on the other hand, the answer is the sum over all possible values of k, of the number of subcommittees consisting of k Democrats and r − k Republicans.
Generalized Vandermonde's identity
If in the algebraic derivation above more than two polynomials are used, it results in the generalized Vandermonde's identity. For y + 1 polynomials:
The hypergeometric probability distribution
When both sides have been divided by the expression on the left, so that the sum is 1, then the terms of the sum may be interpreted as probabilities. The resulting probability distribution is the hypergeometric distribution. That is the probability distribution of the number of red marbles in r draws without replacement from an urn containing n red and m blue marbles.
Chu–Vandermonde identity
The identity generalizes to noninteger arguments. In this case, it is known as the Chu–Vandermonde identity (see Askey 1975, pp. 59–60) and takes the form
for general complexvalued s and t and any nonnegative integer n. This identity may be rewritten in terms of the falling Pochhammer symbols as
in which form it is clearly recognizable as an umbral variant of the binomial theorem. (For more on umbral variants of the binomial theorem, see binomial type) The Chu–Vandermonde identity can also be seen to be a special case of Gauss's hypergeometric theorem, which states that
where is the hypergeometric function and Γ(n + 1) = n! is the gamma function. One regains the Chu–Vandermonde identity by taking a = −n and applying the identity
liberally.
References
 Askey, Richard (1975), Orthogonal polynomials and special functions, Regional Conference Series in Applied Mathematics, 21, Philadelphia, PA: SIAM, pp. viii+110
Categories: Factorial and binomial topics
 Mathematical identities
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