$\displaystyle F_{k}(\nu)=(\nu^{2}-1^{2})(\nu^{2}-3^{2})\cdots(\nu^{2}-(2k-1)^{%
2})=(-4)^{k}{\left(\frac{1}{2}-\frac{\nu}{2}\right)_{k}}{\left(\frac{1}{2}+%
\frac{\nu}{2}\right)_{k}},$