Given the following recursive relation:
$a_0 = 1,$ $a_n = a_{n-1}(p-2q)+2(-p)^n$
is there a simple function that has this as its Taylor series, i.e.
$f(x) = \sum_{n=0}^{\infty} \frac{a_n}{n!}x^n$
where $f$ is something (relatively) simple?
Note: This function is derived from $f(x) = (pe^{-px} - qe^{(p-2q)x})/(p-q)$, which surprisingly has no variables $p$ or $q$ in its denominator.
Thanks!