Lagrange notation: $f^{(0)}(x)$?

Question

Using Lagrange notation, is $f^{(0)}(x)=f(x)$? Is this standard notation, or would one have to define $f^{(0)}(x)=f(x)$ first, before using it? Aside: the context of the question is whether to include the first term within the summation when expressing the Taylor series and hence start at $n=0$, or to write it separately outside the summation and start the summation at $n=1$. The former has been done here: https://en.wikipedia.org/wiki/Taylor_series#Definition

Answer 1

Writing $f^{(0)}(x)=f(x)$ just to be sure that we all agree on the same definitions is always smart to do. When in doubt specify unless it might even make things more confusing. Interpreting a function as its own zeroth derivative is not a weird thing to think about, but just mention it in case the reader is very fussy and wants to make a point about everything (knowing myself - I would). I think your reasoning is valid.

Answer 2

A definition of something is always good before using it, as happend in the wikipedia article:

The derivative of order zero of $f$ is defined to be $f$ itself.

So you can start summation at $n=0$ in Taylor series after refering to this definition. But you can put your mind at rest. Most mathematicians would expect that $f^{(0)}(x)=f(x)$ without definition.