It is well known that:
$$\sum_{n=1}^\infty \frac{n q^n}{1-q^n} = -q \frac{d}{dq} \Phi(q)$$
Where $\Phi$ is defined as follows for $|q|<1$:
$$\Phi(q) = \prod_{n=1}^\infty (1-q^n)$$
- Is there any similar expression that showcases any type of relationship between $\Phi(q)$ and:
$$\sum_{n=1}^\infty \frac{q^n}{1-q^n}$$
?
- Is there any related result (expressing the Lamber series in terms of $\Phi$) for an integer $\alpha > 1$ in:
$$\sum_{n=1}^\infty \frac{n^\alpha q^n}{1-q^n}$$
?
Thank you in advance.