Chu-Vandermonde Identity/Rising Factorial Variant

Theorem

Let $r, s \in \R, n \in \Z_{\ge 0}$.

Then:

$\ds \sum_{k \mathop = 0}^n \dbinom n k r^{\overline k} s^{\overline {n-k} } = \paren {r + s}^{\overline n}$


Proof

From Rising Factorial as Factorial by Binomial Coefficient, we have:

\(\ds r^{\overline k}\) \(=\) \(\ds k! \dbinom {r + k - 1} k\)
\(\ds s^{\overline {n - k} }\) \(=\) \(\ds \paren{n - k}! \dbinom {s + n - k - 1} {n - k}\)
\(\ds \paren{r + s}^{\overline n}\) \(=\) \(\ds n! \dbinom {r + s + n - 1} n\)

Therefore:

\(\ds \sum_{k \mathop = 0}^n \dbinom n k r^{\overline k} s^{\overline {n - k} }\) \(=\) \(\ds \sum_{k \mathop = 0}^n \paren {\dfrac {n!} {k! \paren{n - k}!} } \paren{ {k! \dbinom {r + k - 1} k} } \paren{ {\paren{n - k}! \dbinom {s + n - k - 1} {n - k} } }\) Definition of Binomial Coefficient
\(\ds \) \(=\) \(\ds n! \sum_{k \mathop = 0}^n {\dbinom {r + k - 1} k} \dbinom {s + n - k - 1} {n - k}\) $k!$ and $\paren{n - k}!$ cancel
\(\ds \) \(=\) \(\ds n! \sum_{k \mathop = 0}^n {\paren{-1}^k \dbinom {-r} k} \paren{-1}^{n - k} \dbinom {-s} {n - k}\) Negated Upper Index of Binomial Coefficient/Corollary 2
\(\ds \) \(=\) \(\ds {\paren{-1}^n n! \sum_{k \mathop = 0}^n \dbinom {-r} k} \dbinom {-s} {n - k}\) Product of Powers $\paren{-1}^k \times \paren{-1}^{n - k} = \paren{-1}^n$
\(\ds \) \(=\) \(\ds \paren{-1}^n n! \binom {-\paren{r + s} } n\) Chu-Vandermonde Identity
\(\ds \) \(=\) \(\ds n! \dbinom {r + s + n - 1} n\) Negated Upper Index of Binomial Coefficient
\(\ds \) \(=\) \(\ds \paren{r + s}^{\overline n}\) Rising Factorial as Factorial by Binomial Coefficient

$\blacksquare$


Also known as

When $r$ and $s$ are integers, the Chu-Vandermonde identity is more commonly known as Vandermonde's identity, Vandermonde's convolution (formula) or Vandermonde's convolution.

Sometimes it is seen referred to as the Chu-Vandermonde formula, or Vandermonde's theorem.


Source of Name

This entry was named for Chu Shih-chieh and Alexandre-Théophile Vandermonde.


Sources

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