How can one write $e^{a(x) \cdot b(x)} = c(x) e^{ b(x) }$ with $c(x)$ not implicitly depending on $b(x)$. I do not believe this is generally possible so alternatively one can use an infinite series or infinite product $\sum c_k x^k e^{ d_k \cdot b(x)}$ or $\prod e^{c_k \cdot x^k} e^{d_k \cdot b(x)} $
Maybe there is another way using the Lambert $W$ function or some other special function. I'm looking for a quasi power to product formula for exp. e.g., $e^{a \cdot b} = e^{a^b} = Q(a) \cdot e^b$.
Obviously one can write $e^{a b} = e^{(a - 1)b} e^b$ but I'm looking to remove the direct dependence on one factor. Here I'm dealing with functions and the requirements are slightly relaxed. I.e., take the infinite sum and product cases as the main interpretation and not the examples involving constants... which do not have solutions in general.
for example, using the power series exapansion of $a(x)$ we end up with
$e^{b(x)\sum a_k x^k} = \prod e^{a_k x^k b(x)}$
but I need to get this to look something like
$\prod Q(x) e^{a_k b(x)}$
without $Q(x)$ depending implicitly on $b(x)$ (else I would just go for simpler case $e^{(a(x) - 1) b(x)} e^{b(x)}$.
(the fact that we still allow for dependence between the factors through $a_k$ makes me think that such an expansion is possible but maybe this is wishful thinking...)