Kummer's function

In mathematics, Kummer's function, named after Ernst Kummer, is a complex function related to the polylogarithm. It is defined by^[1]

${\displaystyle \Lambda _{n}(z)=\int _{0}^{z}{\frac {\log ^{n-1}|t|}{1+t}}\;dt.}$

Its duplication formula is

${\displaystyle \Lambda _{n}(z)+\Lambda _{n}(-z)=2^{1-n}\Lambda _{n}(-z^{2})}$.

Compare this to the duplication formula for the polylogarithm:

${\displaystyle \operatorname {Li} _{n}(z)+\operatorname {Li} _{n}(-z)=2^{1-n}\operatorname {Li} _{n}(z^{2}).}$

An explicit link to the polylogarithm is given by

${\displaystyle \operatorname {Li} _{n}(z)=\operatorname {Li} _{n}(1)\;\;+\;\;\sum _{k=1}^{n-1}(-1)^{k-1}\;{\frac {\log ^{k}|z|}{k!}}\;\operatorname {Li} _{n-k}(z)\;\;+\;\;{\frac {(-1)^{n-1}}{(n-1)!}}\;\left[\Lambda _{n}(-1)-\Lambda _{n}(-z)\right].}$