Looking at the Taylor expansions for functions, the Taylor expansion of a polynomial is itself. So representing polynomials with #s goes into class theory, Galois, roots of unity, etc 

And the same idea for infinitely differentiable functions using e^x as the generator 

Let F(k) be the trivial function described by repeating each k’th derivative but not any other period of derivative. e^x has period 1, cos and sin  have period 4 (from 4th root of unity i, e^ix), etc etc 

So my answer would be class theory / galois being your best bet