Suppose that for a convergent series $\displaystyle f(x) = \sum_{n=0}^\infty c_n \cdot x^n $ we form a new (sub)series $\displaystyle f_p(x) = \sum_{n=0, n\in \mathbb{P}}^\infty c_n \cdot x^n $ That means that we keep just coefficients for prime indices. For instance, $\text{exp}_p(x)=\cfrac{x^2}{2!} + \cfrac{x^3}{3!} + \cfrac{x^5}{5!} + \cfrac{x^7}{7!} + \cfrac{x^{11}}{11!} + \cdots$.
My question(s) are very general since I am not familiar with the topic (and I am not active in mathematics anymore). I am interested in properties of the transformation $f\mapsto f_p$.
- Has anyone studied this transformation?
- What are its properties?
- Could such transformations be useful (having real world application)?
- What are the transformations of known functions, like $\exp(x), \sin(x), \ldots$?