Euler function

:"For other meanings, see List of topics named after Leonhard Euler".In mathematics, the Euler function is given by:$phi(q)=prod\_\{k=1\}^infty\; (1-q^k).$Named after Leonhard Euler, it is a prototypical example of a q-series, a modular form, and provides the prototypical example of a relation between combinatorics and complex analysis.

Properties

The coefficient $p(k)$ in the Maclaurin series for $1/phi(q)$ gives the number of all partitions of k. That is,:$frac\{1\}\{phi(q)\}=sum\_\{k=0\}^infty\; p(k)\; q^k$where $p(k)$ is the partition function of k.

The Euler identity is:$phi(q)=sum\_\{n=-infty\}^infty\; (-1)^n\; q^\{(3n^2-n)/2\}.$

Note that $(3n^2-n)/2$ is a pentagonal number.

The Euler function is related to the Dedekind eta function through a Ramanujan identity as :$phi(q)=\; q^\{-1/24\}\; eta(\; au)$

where $q=e^\{2pi\; i\; au\}$ is the square of the nome.

Note that both functions have the symmetry of the modular group.

References

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