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.
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$\begingroup$ I made the mistake initially of assuming the infinite series $S(a,b)$ converges for all $a>0,\,b>0$, but it seems like it only converges for $a<2b$. Does that sound right to you? $\endgroup$– David HCommented Jul 23, 2023 at 17:11
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