Lerch transcendent

In mathematics, the Lerch transcendent, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about a similar function in 1887.^[1] The Lerch transcendent, is given by:

${\displaystyle \Phi (z,s,\alpha )=\sum _{n=0}^{\infty }{\frac {z^{n}}{(n+\alpha )^{s}}}}$.

It only converges for any real number ${\displaystyle \alpha >0}$, where ${\displaystyle |z|<1}$, or ${\displaystyle {\mathfrak {R}}(s)>1}$, and ${\displaystyle |z|=1}$.^[2]