Is the notion of elementary function a fluid concept among algebras?

[Anixx]

TL;DR Summary

Is it possible that in certain rings the power series representing special functions are expressable via series representing elementary functions?

Let's consider the Taylor power series of a function on real numbers.

Some of them represent elementary functions, and some of them represent special functions. The special functions cannot be expressed via finite combination of elementary functions on real or complex numbers.

Now, take some different ring (commutative of anticommutative). Is it possible that the power series, representing special functions on real numbers, can be represented as finite combination of the series, representing elementary functions in that ring?

The exponentiation, multiplication and addition operations in the power series expansions should be taken from that ring which we examine, while the power series themselves should be identical to those on reals.

For instance, sine function cannot be expressed via exponentiation on real numbers but can be on complex numbers due to algebraic properties of the imaginary unit. But this function is defined elementary anyway. What about such functions as digamma, gamma, zeta? Can their power series be expressed via the powerseries, corresponding to elementary functions in some rings?

What if we add a condition that the ring should include real numbers? Or, at least, integers?

 

[fresh_42]

The answer you should get here is 'No'.

However, if we instead clean up your confusion, it might take a while. There is no such thing as a special or an elementary function. Next, what should a change of rings do? E.g. you could consider the epimorphism along an ideal $x^n$ and the power series collapses to a polynomial. The same can happen to the coefficients if we make them cyclic. I'm afraid your question is far too vague to give a reasonable answer, hence: no.

 

[Delta2]

I think changing the basic field or ring of our algebra might make some functions that are elementary in one, not elementary in the other (your example of the $\sin,\cos$ functions as we change R with C, they are elementary in C but not elementary in R) but i think there will always be new functions that are not elementary in the new field or new ring.