Nielsen’s Generalized Polylogarithms

Properties (in particular functional relations and special values) of the functions \[{gathered} ( - 1)n + p - 1 (n - 1)!p!Sn,p (z) = ∫_0^1 {log n - 1 tlog ^p (1 - zt){{dt}} {t}} , \\ ( - 1)n + p - 1 (n - 1)!p!Ln,p (z) = ∫_0^z {log n - 1 tlog ^p (1 - t){{dt}} {t}} , \\ ( - 1)n + p - 1 (n - 1)!p!Mn,p (z) = ∫_0^z {log n - 1 tlog ^p (1 + t){{dt}} {t}} , \\ {gathered} \] 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.