Do there exist closed form functional representations for the following products?: $$ f(x,\alpha) = \prod_{n=1}^\infty \left( 1 + \alpha (n) \ x^n \right) $$ Where $\alpha$ is some multiplicative function.
As examples:
For $\alpha(n) = 1$, this reduces to a form of the q-pochhammer symbol.
Expansion in terms of the elementary symmetric polynomials may be obtained.
Any of the following insights would be appreciated:
- Are there any other simple cases for common multiplicative functions; such the Möbius function or any of the other classical functions?
- Do any references exist for this type of product in general?
- Are there any 'nice' series representations?
- What properties might such a closed form hold?
Edit:
Questions of covergence can be readily seen as this convergence is equivalent to the convergence of:
$$
\sum_{n=1}^\infty \alpha(n) x^n
$$