Special functions Hypergeometric functions Theorem

Chu–Vandermonde identity

Builds on Binomial coefficient Pochhammer symbol Hypergeometric series

Statement

Let $n \ge 0$ be an integer.

For all $x, y \in \C$ , $\binom{x + y}{n} = \sum_{k=0}^{n} \binom{x}{k} \binom{y}{n - k},$ where $\binom{x}{k} = \frac{x(x-1)\cdots(x-k+1)}{k!}$ is the binomial coefficient.
Equivalently, for all $b, c \in \C$ with $c \notin \set{0, -1, -2, \ldots}$ , ${}_2F_1(-n, b; c; 1) = \frac{(c - b)_n}{(c)_n},$ where ${}_2F_1$ is the terminating hypergeometric series and $(x)_n$ is the Pochhammer symbol.

Remarks

Both sides of part 1 are polynomials in $x$ and $y$ . For natural numbers $x = m$ and $y = p$ the identity is Vandermonde's identity, proved by counting $n$ -element subsets of a union of an $m$ -set and a $p$ -set. A polynomial in two variables that vanishes on $\N \times \N$ is zero, so part 1 holds for all $x, y$ . Part 2 is also the case $a = -n$ of Gauss's hypergeometric theorem, since $\frac{\Gamma(c) \Gamma(c + n - b)}{\Gamma(c + n) \Gamma(c - b)} = \frac{(c - b)_n}{(c)_n}$ .

Example: with $x = y = \frac12$ and $n = 2$ , the left side is $\binom{1}{2} = 0$ , and the right side is $\binom{1/2}{2} + \binom{1/2}{1}^2 + \binom{1/2}{2} = -\frac18 + \frac14 - \frac18 = 0$ .

The name records that the identity was known to Zhu Shijie long before Vandermonde.

Going deeper

For a student

In hypergeometric language it is the summation of a terminating ${}_2F_1$ at $z = 1$ . It is the simplest case of a whole hierarchy of terminating summations, followed by the Pfaff–Saalschütz formula for ${}_3F_2$ and Dougall's formula. Its $q$ -analogue is the q-Vandermonde identity.

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Tag	`4YK46R1QDR`
URI	`m2://4YK46R1QDR`
Kind	statement, theorem
Writing	W0 stub
MSC	33C05, 05A19
Status	This copy of the site has no build data. Run `m2 status 4YK46R1QDR`.
About	Hypergeometric series
Uses	Binomial coefficient, Pochhammer symbol, Hypergeometric series