Does the following series have a solution in terms of standard-ish funcitons (e.g. hypergeometric)? If not, does it at least have a name, or papers discussing it? (It seems so simple that it ought to have been studied before.)
The series is given by:
$$f(x,\rho)=\sum_{k=1}^\infty{\frac{x^k}{k!(1-\rho^k)}}$$
where $|\rho|<1$.
Series of roughly this form arise when considering the cumulant generating function of autoregressive processes of order 1.
Clearly $f(x,0)=\exp{x}-1$, but I do not know much more than that.
Plotting $f(x,\rho)-\exp{x}+1$ shows a moderately convex function with derivative around $1$ at $0$. Maple fails to give a series solution for $f$ in $\rho$ about $\rho=0$, but about $\rho=1$ it gives:
$$f(x,\rho)=\frac{g(x)}{1-\rho} + \frac{1}{2} \left( \exp{x}-1-g(x) \right) + \frac{1}{12} \left( x \exp{x} - g(x) \right) (1-\rho) + \frac{1}{24} \left( x \exp{x} - g(x) \right) (1-\rho)^2 +O((1-\rho)^3).$$
where:
$$g(x)= - \gamma - \log(-x) - \operatorname{Ei}_1(-x).$$
(The third order term is given by a messy hypergeometric function.)
I do not know how much this could help
Suppose $0<\rho <1$ and write $$\frac 1{1-\rho^k}=\sum_{n=0}^\infty \rho^{kn}$$ $$f(x,\rho)=\sum_{k=\color{red}{0}}^\infty{\frac{x^k}{k!\,(1-\rho^k)}}=\sum_{n=0}^\infty \exp\big[x \, \rho^n\big]$$ which is the sum of exponential functions with smaller and smaller arguments.