Chu-Vandermonde Identity

$$(x+a)_n=\sum_{k=0}^\infty {n\choose k}(a)_k(x)_{n-k}$$

where ${n\choose k}$ is a Binomial Coefficient and $(a)_n\equiv a(a-1)\cdots(a-n+1)$ is the Pochhammer Symbol. A special case gives the identity

$$\sum_{l=0}^{\max(k,n)} {m\choose k-l}{n\choose l} = {m+n\choose k}.$$

See also Binomial Theorem, Umbral Calculus

References

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, pp. 130 and 181-182, 1996.