$\sum_{i=0}^{S/2} (S/2+3/2)^{2i}\frac{2\cdot\Gamma(a) \Gamma(i+0.5)}{\sqrt(\pi) \Gamma(a+i+1)} {S \choose 2i} B_{\frac{S-2i}{2}}^{1+S}(0.5+S/2) $

Question

Show that the following sum \begin{equation} K=\sum\limits_{i=0}^{S/2} (\frac{S}{2}+3/2)^{2i}\frac{2\cdot\Gamma(\alpha) \Gamma(i+0.5)}{\sqrt(\pi) \Gamma(\alpha + i +1)} {S \choose 2i} B_{\frac{S-2i}{2}}^{1+S}(0.5+\frac{S}{2}) \end{equation} is positive for any even integer S and $\alpha>0.5, \alpha < 12$. $B_{j}^{k}$ is the generalized Bernoulli polynomial. I can only check this numerically, because this is an alternating sum, but maybe anyone has an idea how to show this in general?