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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 t 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?

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  • $\begingroup$ Take the ring consisting two elements $0,1$. $\endgroup$
    – markvs
    Jan 4, 2021 at 21:13
  • $\begingroup$ @dodd do you mean something like boolean algebra or p-adics? This would be a good answer, but what if we add a condition that the ring should include real numbers? Or, at least, integers? $\endgroup$
    – Anixx
    Jan 4, 2021 at 21:16
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    $\begingroup$ How exactly do you define "elementary" or "special" functions over a general ring? What about "finite combinations"? $\endgroup$
    – Somos
    Jan 5, 2021 at 20:07
  • $\begingroup$ @Somos I am asking, whether the functions that are not elementary on reals (can not be expressed via trigonometric, inverse trigonometric, logarithms, exponentials and arithmetic operations in closed form) can be elementary on another ring (that is can be expressed via those functions). $\endgroup$
    – Anixx
    Jan 5, 2021 at 20:14

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