A closed form of $\sum_{k=1}^\infty \psi^{(1)} (k+a)\psi^{(1)} (k+b)$?

Question

The following result $$ \sum_{k=1}^\infty\left(\psi^{(1)} (k)\right)^2 = 3\zeta(3) $$ where $\psi^{(1)}$ is the polygamma function makes me think there is a nice sum for the series

$$ \sum_{k=1}^\infty \left(\psi^{(1)} (k+a)\right)^2 $$

or

$$ \sum_{k=1}^\infty \psi^{(1)} (k+a)\psi^{(1)} (k+b) $$

where $a$ and $b$ are any real numbers such that $a >-1, b>-1.$

Could you help me to find it?