Chu-Vandermonde Identity/Extended

Theorem

Let $r, s, \alpha, \beta \in \C$ be complex numbers.

Then:

$\ds \sum_{k \mathop \in \Z} \dbinom r {\alpha + k} \dbinom s {\beta - k} = \dbinom {r + s} {\alpha + \beta}$

where $\dbinom r {\alpha + k}$ denotes a binomial coefficient.

Proof

From the Chu-Vandermonde Identity, we have:

$\ds \sum_{k \mathop \in \Z} \binom r k \binom s {n - k} = \binom {r + s} n$

Let $n = \alpha + \beta$

Let $k = \alpha + k$

Then:

$\ds \sum_{k \mathop \in \Z} \binom r {\alpha + k} \binom s {\alpha + \beta - \paren {\alpha + k} } = \binom {r + s} {\alpha + \beta}$

$\ds \sum_{k \mathop \in \Z} \binom r {\alpha + k} \binom s {\beta - k } = \binom {r + s} {\alpha + \beta}$

$\blacksquare$

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Sources

• 1977: Lyle Ramshaw: Binomial coefficients with non-integral lower index (Inf. Proc. Letters Vol. 6: pp. 223 – 226)
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $42$ (Solution)