Sum over k of m+r+s Choose k by n+r-s Choose n-k by r+k Choose m+n
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Theorem
Let $m, n \in \Z_{\ge 0}$.
Then:
- $\ds \sum_{k \mathop \in \Z} \binom {m - r + s} k \binom {n + r - s} {n - k} \binom {r + k} {m + n} = \binom r m \binom s n$
Proof
\(\ds \) | \(\) | \(\ds \sum_{k \mathop \in \Z} \binom {m - r + s} k \binom {n + r - s} {n - k} \binom {r + k} {m + n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \in \Z} \binom {m - r + s} k \binom {n + r - s} {n - k} \paren { \sum_{j \mathop \in \Z} \binom r {m + n - j} \binom k j}\) | Chu-Vandermonde Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \in \Z} \sum_{j \mathop \in \Z} \paren {\binom {m - r + s} k \binom k j } \binom {n + r - s} {n - k} \binom r {m + n - j}\) | rearranging | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \in \Z} \sum_{j \mathop \in \Z} \binom {m - r + s} j \binom {m - r + s - j} {k - j} \binom {n + r - s} {n - k} \binom r {m + n - j}\) | Product of $\dbinom r m$ with $\dbinom m k$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop \in \Z} \binom {m - r + s} j \binom r {m + n - j} \paren {\sum_{k \mathop \in \Z} \binom {m - r + s - j} {k - j} \binom {n + r - s} {n - k} }\) | grouping | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop \in \Z} \binom {m - r + s} j \binom r {m + n - j} \binom {m + n - j} {n - j}\) | Chu-Vandermonde Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop \in \Z} \binom {m - r + s} j \binom r {m + n - j} \binom {m + n - j} m\) | Symmetry Rule for Binomial Coefficients | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop \in \Z} \binom {m - r + s} j \binom r m \binom {r - m} {n - j}\) | Product of $\dbinom r m$ with $\dbinom m k$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \binom r m \paren {\sum_{j \mathop \in \Z} \binom {m - r + s} j \binom {r - m} {n - j} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \binom r m \binom s n\) | Chu-Vandermonde Identity |
$\blacksquare$
Also see
Sources
- 1797: Johann Friedrich Pfaff: Observationes analyticae ad L. Euleri Institutiones calculi integralis (Nova Acta Acad. Scient. Petr. Vol. 11: pp. 38 – 57)
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $31$