Evaluate $\sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(2b)_n} \frac{\psi^{(0)}\left (n+ \frac{a}2+1 \right ) }{\Gamma\left (n+ \frac{a}2+1 \right ) }$

Question

Define $$S(a,b)=\sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(2b)_n} \frac{\psi^{(0)}\left (n+ \frac{a}2+1 \right ) }{\Gamma\left (n+ \frac{a}2+1 \right ) },$$ where $\psi^{(0)}(x)=\frac{\Gamma^\prime(x)}{\Gamma(x)}$ is digamma function and $(a)_n=\frac{\Gamma(n+a)}{\Gamma(a)}$ is the Pochhammer-symbol. Can we evaluate it in a closed-form(at least for certain $a,b$)?
I was motivated by the relation between $$ \,_4F_3\left ( \frac{a}{2},\frac{a}2,a,b;a-b+1,1+\frac{a}2,1+\frac{a}{2} ;1\right ) $$ and $S(a,b)$. Actually(see explanations), $$ S\left ( \frac23,\frac23 \right )=\frac{\left ( -18\gamma+13\pi\sqrt{3}-27\ln(3) \right )\Gamma\left ( \frac13 \right )^8 }{192\pi^4}. $$ As a corollary, we have $$ \,_4F_3\left ( \frac13,\frac13,\frac23,\frac23; 1,\frac43,\frac43;1 \right ) =\frac{\Gamma\left ( \frac13 \right )^6 }{36\pi^2}. $$ If more $S(a,b)$ are possible, there must be some corresponding expressions for ${}_4F_3$s. Any help will be appreciated.