I am trying to follow a proof in a physics paper, but got stuck with the identity
$$\sum_{i=0}^n(-1)^i\binom{k}{n-i}\frac{(m+i)!}{i!} = m!\binom{k-m-1}{n}.$$
I would be very grateful if you could shed light on this mystery.
Alternating sum over binomial coefficients
binomial-coefficientssummation
Best Answer
Here we have Chu-Vandermonde's Identity in disguise.
Comment:
In (1) we use the binomial identity $\binom{-p}{q}=\binom{p+q-1}{q}(-1)^q$.
In (2) we apply the Chu-Vandermonde identity.