Summation involving Binomial coefficients

Question

$$S(m,k)=\sum_{n=m}^{k}(-1)^n\binom{n}{m}$$ Is possible to get closed form solution of this sum? We know $$T(m,k)=\sum_{n=m}^{k}\binom{n}{m}=\binom{k+1}{m+1}$$ But what if the sign alternates?

Answer 1

There is an expression but is not a closed form $$S(m,k)=\sum_{n=m}^{k}(-1)^n\binom{n}{m}=(-1)^k \binom{k+1}{m} \, _2F_1(1,k+2;k+2-m;-1)+(-1)^m\, 2^{-(m+1)}$$ where appears the Gaussian or ordinary hypergeometric function.