Nielsen’s Generalized Polylogarithms

Source abstract

Properties (in particular functional relations and special values) of the functions \[\begin{gathered} ( - 1)^{n + p - 1} (n - 1)!p!S_{n,p} (z) = \int_0^1 {\log ^{n - 1} t\log ^p (1 - zt)\frac{{dt}} {t}} , \hfill \\ ( - 1)^{n + p - 1} (n - 1)!p!L_{n,p} (z) = \int_0^z {\log ^{n - 1} t\log ^p (1 - t)\frac{{dt}} {t}} , \hfill \\ ( - 1)^{n + p - 1} (n - 1)!p!M_{n,p} (z) = \int_0^z {\log ^{n - 1} t\log ^p (1 + t)\frac{{dt}} {t}} , \hfill \\ \end{gathered} \] which play a role in the computation of higher order radiative corrections in quantum electrodynamics, are discussed for complex z and positive integers n and p. The first function is a generalization of the well-known polylogarithms $(p = 1)$. The discussion is based on results published by Nielsen early this century in a little-known monograph.