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Zbl 0606.33013
Kölbig, K.S.
Nielsen's generalized polylogarithms. (English)
[J] SIAM J. Math. Anal. 17, 1232-1258 (1986). ISSN 0036-1410; ISSN 1095-7154/e

Properties (in particular functional relations and special values) of the functions $$ (-1)\sp{n+p-1}(n-1)! p! S\sb{n,p}(z)=\int\sp{1}\sb{0}\log\sp{n-1}t \log\sp p(1-zt)(dt/t), $$ $$ (-1)\sp{n+p-1}(n-1)! p! L\sb{n,p}(z)=\int\sp{z}\sb{0}\log\sp{n-1}t \log\sp p(1-t)(dt/t), $$ $$ (-1)\sp{n-1}(n-1)! p! M\sb{n,p}(z)=\int\sp{z}\sb{0}\log\sp{n-1}t \log\sp p(1+t)(dt/t), $$ 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.

MSC 2000:: *33E99 Special functions

Keywords: Nielsen's generalized polylogarithms; Spence functions; logarithmic integrals; Riemann zeta functions; Stirling numbers of the first kind

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