Resurgence of the fractional polylogarithms (Q971478)

For a complex number \(\alpha\), let \[ \text{Li}_\alpha(z):=\sum_{n=1}^\infty \frac{z^n}{n^\alpha} \] be the \(\alpha\)-polylogarithm function. The authors show that \(\text{Li}_\alpha(z)\) is a multivalued analytic function in the complex plane minus 0 and 1 and prove the analytic continuation for non-integer values of \(\alpha\). Two corollaries of their main result are the following. Firstly, for \(\alpha\) such that \(\text{Re}\,\alpha< 0\) and \(z\) such that \(\text{Re}\, z < 0\) and \(|z| < 2\pi\), they show that \[ Li_\alpha(e^z):=e^{-\pi i(\alpha+1)}\Gamma(1-\alpha)z^{\alpha-1}+\sum_{n=0}^\infty\frac{\zeta(\alpha-n)}{n!}z^n, \] where \(\zeta\) is the Riemann zeta-function. Secondly, for \(\text{Re}\,\alpha > 0\) and \(z\) large, they show that \[ \text{Li}_\alpha(z)=\frac{1}{\Gamma(\alpha+1)}((\log z)^\alpha+o(1)). \]