Finitely differentiable functions

Functions that have a differentiation iteration type with a finite size are solutions to a simple two element homogeneous LODE with constant coefficients. The solutions to a LODE can be described by a basis which in this case can be ordered by the total ordering of rational number sequences.

The function $x^2+2x+1$ would become [1,2,1,0] for the LODE $y''''=y'''$. If we set the default value of each field in this ordered basis this vector could be reduced to [1,2,1] which is equivalent to description of this function as a polynomial. The function $e^x$ is unaffected by differentiation so it would simply be [1] for $y'=y$.

The hyperbolic trigonometric functions $cosh(x)$ and $sinh(x)$ are involutions under differentiation so they would be represented by [1/2,1/2] and [1/2,-1/2]. The trigonometric functions $sin(x)$ and $cos(x)$ have order four so they would be represented as [0,0,1,0] and [0,0,0,1].