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Concept
Euler function
Summary
In mathematics, the Euler function is given by
Named after Leonhard Euler, it is a model example of a q-series and provides the prototypical example of a relation between combinatorics and complex analysis.
The coefficient in the formal power series expansion for gives the number of partitions of k. That is,
where is the partition function.
The Euler identity, also known as the Pentagonal number theorem, is
is a pentagonal number.
The Euler function is related to the Dedekind eta function as
The Euler function may be expressed as a q-Pochhammer symbol:
The logarithm of the Euler function is the sum of the logarithms in the product expression, each of which may be expanded about q = 0, yielding
which is a Lambert series with coefficients -1/n. The logarithm of the Euler function may therefore be expressed as
where -[1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, 15/8, 13/9, 18/10, ...
Official source
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