Nielsen’s Generalized Polylogarithms

Properties (in particular functional relations and special values) of the functions (- 1) ^n + p - 1 (n - 1) !p!S₍, ₏ (z) = ₀¹ ^{n - 1 t ᵖ (1 - zt) dt t}, \\ (- 1) ^n + p - 1 (n - 1) !p!L₍, ₏ (z) = ₀ᶻ ^{n - 1 t ᵖ (1 - t) dt t}, \\ (- 1) ^n + p - 1 (n - 1) !p!M₍, ₏ (z) = ₀ᶻ ^{n - 1 t 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.