March – A representation of the generalized hypergeometric function $_3F_2$

Here is a representation of the hypergeometric function $_3F_2(1,1,b;2,2;\cdot)$ in terms of the polylogarithm: For $\vert x\vert<1$ and $b\in\mathbb{N}$, the hypergeometric series representation is given by

$$_3F_2(1,1,1+b;2,2;x)= \sum^{\infty}_{n=0}\frac{(1)_n(1)_n(1+b)_n}{(2)_n(2)_n} \frac{x^n}{n!} \nonumber$$

$$=\sum^{\infty}_{n=0}\frac{(1+b)_n}{(n+1)^2n!}x^n = \frac{1}{b!}\sum^{\infty}_{n=0} \frac{(n+1)_{b}}{(n+1)^2}x^n \nonumber$$

$$\overset{(a)}{=} \frac{1}{b!} \sum^{\infty}_{n=0} \frac{\sum^{b}_{k=1}\left[ b \atop k \right](n+1)^k}{(n+1)^2} x^n\nonumber$$

$$= \frac{1}{b!} \sum^{b}_{k=1}\left[ b \atop k \right] \sum^{\inft... ...frac{1}{b!}\sum^{b}_{k=1} \left[ b \atop k \right]\frac{\text{Li}_{2-k}(x)}{x},$$

where $\left[ b \atop k \right]$ is the unsigned Stirling number of the first kind. In (a), we used the expansion of the rising Pochhammer factorial and in (b), we used the definition of the polylogarithm. Furthermore, the $x \in \mathbb{C}$ follows from the analytic continuation of the polylogarithm.