Euler function

In mathematics, the Euler function is given by

\phi (q)=\prod _{k=1}^{\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 {1}{\phi (q)}}=\sum _{k=0}^{\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 _{n=-\infty }^{\infty }(-1)^{n}q^{(3n^{2}-n)/2}.

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

The Euler function is related to the Dedekind eta function as

\phi (e^{2\pi i\tau })=e^{-\pi i\tau /12}\eta (\tau ).

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

\phi (q)=(q;q)_{\infty }.

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 _{n=1}^{\infty }{\frac {1}{n}}\,{\frac {q^{n}}{1-q^{n}}},

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

\ln(\phi (q))=\sum _{n=1}^{\infty }b_{n}q^{n}

where $b_{n}=-\sum _{d|n}{\frac {1}{d}}=$ -[1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, 15/8, 13/9, 18/10, ...] (see OEIS A000203)

On account of the identity $\sigma (n)=\sum _{d|n}d=\sum _{d|n}{\frac {n}{d}}$ , where $\sigma (n)$ is the sum-of-divisors function, this may also be written as

\ln(\phi (q))=-\sum _{n=1}^{\infty }{\frac {\sigma (n)}{n}}\ q^{n}

.

Also if $a,b\in \mathbb {R} ^{+}$ and $ab=\pi ^{2}$ , then^[1]

a^{1/4}e^{-a/12}\phi (e^{-2a})=b^{1/4}e^{-b/12}\phi (e^{-2b}).

Special values

The next identities come from Ramanujan's Notebooks:^[2]

\phi (e^{-\pi })={\frac {e^{\pi /24}\Gamma \left({\frac {1}{4}}\right)}{2^{7/8}\pi ^{3/4}}}

\phi (e^{-2\pi })={\frac {e^{\pi /12}\Gamma \left({\frac {1}{4}}\right)}{2\pi ^{3/4}}}

\phi (e^{-4\pi })={\frac {e^{\pi /6}\Gamma \left({\frac {1}{4}}\right)}{2^{{11}/8}\pi ^{3/4}}}

\phi (e^{-8\pi })={\frac {e^{\pi /3}\Gamma \left({\frac {1}{4}}\right)}{2^{29/16}\pi ^{3/4}}}({\sqrt {2}}-1)^{1/4}

Using the Pentagonal number theorem, exchanging sum and integral, and then invoking complex-analytic methods, one derives^[3]

\int _{0}^{1}\phi (q)\,\mathrm {d} q={\frac {8{\sqrt {\frac {3}{23}}}\pi \sinh \left({\frac {{\sqrt {23}}\pi }{6}}\right)}{2\cosh \left({\frac {{\sqrt {23}}\pi }{3}}\right)-1}}.

References

^ Berndt, B. et al. "The Rogers–Ramanujan Continued Fraction"
^ Berndt, Bruce C. (1998). Ramanujan's Notebooks Part V. Springer. ISBN 978-1-4612-7221-2. p. 326
^ Sloane, N. J. A. (ed.). "Sequence A258232". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001

[1] Berndt, B. et al. "The Rogers–Ramanujan Continued Fraction"

[2] Berndt, Bruce C. (1998). Ramanujan's Notebooks Part V. Springer. ISBN 978-1-4612-7221-2. p. 326

[3] Sloane, N. J. A. (ed.). "Sequence A258232". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

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Euler function

Properties

Special values

References