I have proven the identity $$ \sum_{k=1}^{\infty} \dfrac{\operatorname{_2F_1}(1, 2, 2-1/t,-1/k)}{{k}^{2}} = Γ(2-\dfrac{1}t){\psi^{1/t}(1)}+\psi(-\dfrac{1}t)(\dfrac{1}t(1-\dfrac{1}t))+\gamma(1-\dfrac{1}{t^2} )$$
Which means any fractional derivative of $$\psi(z)$$ at z=1 has a closed form in terms of this summation of hypergeometric functions
I have found 2 special cases with closed forms so far, $$\sum_{k=1}^{\infty} \dfrac{\operatorname{_2F_1}(1, 2, 2-1/2,-1/k)}{{k}^{2}} = \sum_{k=1}^{\infty} \dfrac{\operatorname{arccsch}(\sqrt{k})}{(k+1)^{3/2}} = {\sqrt{\pi}\psi^{1/2}(1)}+\gamma-\operatorname{ln}2 $$ and $$\sum_{k=1}^{\infty} \dfrac{\operatorname{_2F_1}(1, 2, 2-1/4,-1/k)}{{k}^{2}} = \sum_{k=1}^{\infty} \dfrac{{\operatorname{arccoth}({(k+1)}^{1/4})}-\operatorname{arccot}({(k+1)}^{1/4})}{(k+1)^{5/4}} = 2Γ(\dfrac{3}{4})\psi^{1/4}(1)+\dfrac{π}{4}+2\gamma-\dfrac{3}{2}\operatorname{ln}2 $$
I cannot find a closed form for the general case, or closed forms for rational numbers 1/3, 1/5, 1/6, 1/7, 1/8, although I presume they exist.
So my question is is there any other cases that allow for a closed form for the polygamma term without hypergeometric functions? (ie can the sum of hypergeometric functions be expressed in terms of elementary functions like it does in the cases t=2 and t=4) If so, what are they and what are their closed forms in terms of the fractional derivatives of the polygamma function?
This is purely for my curiousity for closed forms. I was intrigued by the idea of putting fractional or even real values for "v" in the polygamma(v,z) function. My goal is to determine a generalized formula to do so, and additionally determine closed forms for the function with a fractional derivative.
I am not sure if the first identity shown at the top has been documented before, but I cannot find any evidence or help with it online.
