Question

are you sure about S=0 for m>0 ? Check for some values please. So you have,

$( (-1)^k \frac{2^{-1+j-k-m} \Gamma[1+m] \Gamma\left[\frac{1}{2}+k+m\right] \Gamma[-j+k+m]}{\Gamma[1+k] \Gamma[1-k+m] \Gamma[1+k+m] \Gamma\left[\frac{1}{2}-j+k+m\right]})$