Chu-Vandermonde Identity/Rising Factorial Variant
Jump to navigation
Jump to search
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
This identity is also known as Vandermonde's formula.
Source of Name
This entry was named for Chu Shih-Chieh and Alexandre-Théophile Vandermonde.
Sources
- Weisstein, Eric W. "Chu-Vandermonde Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Chu-VandermondeIdentity.html