Euler function

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^kwhere 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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