What can be achieved via a partial Euler formula

Question

It is well-known that $$e^x=\sum_{n=0}^\infty\frac{x^n}{n!}.$$ I was curious about the following question: If we, instead of summing over all $n\in\mathbf N$, sum over an infinite proper subset of $\mathbf N$ on the RHS of the Euler formula, what type of function can we obtain? For example, what are $$\sum_{n\equiv 0\,(\text{mod}\,2)}\frac{x^n}{n!},\,\,\,\,\sum_{n\text{ is squared} }\frac{x^n}{n!},\,\,\,\,\sum_{n\text{ is prime}}\frac{x^n}{n!}\,\,?$$ Or can we at least have an estimate on the speed of divergence for cases alike? Are there any known results on this?