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