mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, there are several functions known as Kummer's function. One is known as the

confluent hypergeometric function In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular ...

of Kummer. Another one, defined below, is related to the

polylogarithm In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natur ...

. Both are named for

Ernst Kummer Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of ...

. Kummer's function is defined by :

\Lambda_n(z)=\int_0^z \frac\;dt.

The

duplication formula Duplication, duplicate, and duplicator may refer to: Biology and genetics * Gene duplication, a process which can result in free mutation * Chromosomal duplication, which can cause Bloom and Rett syndrome * Polyploidy, a phenomenon also known ...

is :

\Lambda_n(z)+\Lambda_n(-z)= 2^\Lambda_n(-z^2)

. Compare this to the duplication formula for the polylogarithm: :

\operatorname_n(z)+\operatorname_n(-z)= 2^\operatorname_n(z^2).

An explicit link to the polylogarithm is given by :

\operatorname_n(z)=\operatorname_n(1)\;\;+\;\;
\sum_^ (-)^ \;\frac  \;\operatorname_ (z) \;\;+\;\;
\frac \;\left \Lambda_n(-1) - \Lambda_n(-z) \right

References

*. Special functions {{mathanalysis-stub hu:Kummer-függvény