Kummer's function

Not to be confused with Kummer's confluent hypergeometric function.

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

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

Its duplication formula is

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

Compare this to the duplication formula for the polylogarithm:

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

An explicit link to the polylogarithm is given by

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

References

• Lewin, Leonard, ed. (1991). Structural Properties of Polylogarithms. Mathematical Surveys and Monographs. Vol. 37. Providence, RI: American Mathematical Society. ISBN 0-8218-1634-9.