Given the power series decomposition of a function g, can it be expressed in a "closed form"?

Question

Given the function $ g_{\rho } $ known by its power series/Taylor expansion as:

$ g_{\rho }(x)=\sum _{k=1}^{+\infty } \frac{\left(\rho ^{2^k}\right)^k x^k}{k!}$, with $ 0<\rho<1 $.

We know that its radius of convergence is infinity, and the question is:

can $ g_{\rho } $ be expressed in a "closed form" ?