Euler's Function

In mathematics, the Euler function is given by
:$\phi(q)=\prod_^\infty (1-q^k),\quad , q, <1.$
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.

Properties

The coefficient $p(k)$ in the formal power series expansion for $1/\phi(q)$ gives the number of partitions of ''k''. That is,
:$\frac=\sum_^\infty p(k) q^k$
where $p$ is the partition function.

The Euler identity, also known as the Pentagonal number theorem, is
:$\phi(q)=\sum_^\infty (-1)^n q^.$

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

The Euler function is related to the Dedekind eta function as
:$\phi (e^)= e^ \eta(\tau).$

The Euler function may be expressed as a ''q''-Pochhammer symbol:

:$\phi(q) = (q;q)_.$

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

:$\ln(\phi(q)) = -\sum_^\infty\frac\,\frac,$

which is a Lambert series with coefficients -1/''n''. The logarithm of the Euler function may therefore be expressed as

:$\ln(\phi(q)) = \sum_^\infty b_n q^n$

where $b_n=-\sum_\frac=$ - /1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, 15/8, 13/9, 18/10, ...(see OEISbr>A000203

On account of the identity $\sum_ d = \sum_ \frac,$ this may also be written as

:$\ln(\phi(q)) = -\sum_^\infty \frac \sum_ d.$

Also if $a,b\in\mathbb^+$ and $ab=\pi ^2$, then

:$a^e^\phi (e^)=b^e^\phi (e^).$

Special values

The next identities come from Ramanujan's Notebooks:

: $\phi(e^)=\frac$

: $\phi(e^)=\frac$

: $\phi(e^)=\frac$

: $\phi(e^)=\frac(\sqrt-1)^$

Using the Pentagonal number theorem, exchanging sum and integral, and then invoking complex-analytic methods, one derives

: $\int_0^1\phi(q)\,\mathrmq = \frac.$

References

Notes

Other

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{{Leonhard Euler

Number theory
Q-analogs
Leonhard Euler

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