Chu-Vandermonde Identity/Proof 1
Theorem
| \(\ds \sum_{k \mathop = 0}^n \binom r k \binom s {n - k}\) | \(=\) | \(\ds \binom r 0 \binom s n + \binom r 1 \binom s {n - 1} + \binom r 2 \binom s {n - 2} + \cdots + \binom r n \binom s 0\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \binom {r + s} n\) |
Proof
| \(\ds \sum_{n \mathop = 0}^{r + s} \binom {r + s} n x^n\) | \(=\) | \(\ds \paren {1 + x}^{r + s}\) | Binomial Theorem for Integral Index | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {1 + x}^r \paren {1 + x}^s\) | Exponent Combination Laws | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^r \binom r k x^k \sum_{k \mathop = 0}^s \binom s k x^k\) | Binomial Theorem for Integral Index | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^{r + s} \paren {\sum_{k \mathop = 0}^n \binom r k \binom s {n - k} } x^n\) | Product of Absolutely Convergent Series |
Therefore:
- $\ds \binom {r + s} n = \sum_{k \mathop = 0}^n \binom r k \binom s {n - k}$
$\blacksquare$
Source of Name
This entry was named for Chu Shih-chieh and Alexandre-Théophile Vandermonde.
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-4}$ Generating Functions: Exercise $3$
- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.4$: Factorials and binomial coefficients: $1.35$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $17$