Kummer's function

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

Λ_{n} (z) = \int_{0}^{z} \frac{\log^{n - 1} | t |}{1 + t} d t .

Λ_{n} (z) + Λ_{n} (- z) = 2^{1 - n} Λ_{n} (- z^{2})

.

Compare this to the duplication formula for the polylogarithm:

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

An explicit link to the polylogarithm is given by

{Li}_{n} (z) = {Li}_{n} (1) + \sum_{k = 1}^{n - 1} (- 1)^{k - 1} \frac{\log^{k} | z |}{k!} {Li}_{n - k} (z) + \frac{(- 1)^{n - 1}}{(n - 1)!} [Λ_{n} (- 1) - Λ_{n} (- z)] .

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.

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