I have the following summation
\begin{equation} \frac{(39)_3}{3!} + \frac{(34)_8}{8!}+ \frac{(29)_{13}}{13!}+ \frac{(24)_{18}}{18!} \end{equation}
and i would like to ask if there is any closed form to write it. Observe that the pochhammer symbol is the same as the denominator and each term has a step of 5. So far, I have found that the above can be writen as
\begin{equation} \frac{1}{F_1(-3,38,39,1)} + \frac{1}{F_1(-8,33,34,1)} + \frac{1}{F_1(-13,28,29,1)} + \frac{1}{F_1(-18,23,24,1)} \end{equation} where $F_1$ is the well known hyppergeometric function at $x=1$. I think that i miss an important identity for pochhammer symbols and/or hyppergeometric functions.
$$(a)_b=\frac{\Gamma (a+b)}{\Gamma (a)}$$ $$\frac{(a)_b}{b!}=\frac{\Gamma (a+b)}{\Gamma (a)\, \Gamma (b+1)}$$
So, in you case, the numerator is a constant $\Gamma(?)$. Find it and work the denominators.