Let's consider the function
\begin{align}
\frac{\eta ^m\left( q \right)}{\eta \left( q^m \right)}
\end{align}
here η is the Dedekind eta function $
\eta \left( q \right) =q^{\frac{1}{24}}\prod_{n=1}^{\infty}{\left( 1-q^n \right)}
$,m is a positive integer.
Some special results:
\begin{align}
\frac{\eta ^2\left( q \right)}{\eta \left( q^2 \right)}=\vartheta _4\left( q \right)
\end{align}
here $\vartheta _4\left( q \right)$ is the jacobi theta function
\begin{align}
\frac{\eta ^3\left( q \right)}{\eta \left( q^3 \right)}=\frac{3}{2}a\left( q^3 \right) -\frac{1}{2}a\left( q \right)
\end{align}
here $a\left( q \right)$ is the Borweins' cubic theta function
\begin{align}
a\left( q \right) =1+6\sum_{n=0}^{\infty}{\left\{ \frac{q^{3n+1}}{1-q^{3n+1}}-\frac{q^{3n+2}}{1-q^{3n+2}} \right\}}
\end{align}
and
\begin{align}
\frac{\eta ^5\left( q \right)}{\eta \left( q^5 \right)}=1-5\sum_{n=1}^{\infty}{\left( \frac{n}{5} \right) \frac{nq^n}{1-q^n}}
\end{align}
here $\left( \frac{n}{5} \right) $ denote the Legendre symbol.
Question:Are these kinds of identities universal? Is there a general way to get it?
Asked
Active
Viewed 76 times
3
Loyar
- 89