As I understand it, umbral calculus by default only postulates the effect of evaluation operator on umbral expressions. This means, we are free to add more axioms and relations between umbral elements as long as it does not break the evaluation properties.
Now, consider the expressions
$\exp \left(2 \pi i B_- x\right)$
and
$x \ln \left(\frac{B_+-x}{B_-+x}\right)$,
where $B_-$ is Bernoulli umbra (an umbra with moments equal to Bernoulli numbers) and $B_+=B_-+1$.
The evaluation (scalar part) of these both expressions is equal (if we generalize the powers of $B_-$ via Zeta function $\operatorname{eval} B_-^a=-a \zeta (1-a)$): $$\operatorname{eval} \left(\Re \left(\exp \left(2 \pi i B_- x\right)\right)\right)=\operatorname{eval}\left(x \ln \left(\frac{B_+-x}{B_-+x}\right)\right)= \pi x \cot (\pi x)$$
Its series is
$$1-\frac{\pi ^2 x^2}{3}-\frac{\pi ^4 x^4}{45}-\frac{2 \pi ^6 x^6}{945}-\frac{\pi ^8 x^8}{4725}-\frac{2 \pi ^{10} x^{10}}{93555}-\dots$$
So, seeing the equivalence of the evaluations of the expressons, a question arises, cannot we postulate total equivalence of these two expressions?
$$\exp \left(2 \pi i B_- x\right)\stackrel{?}{=} x \ln \left(\frac{B_+-x}{B_-+x}\right)$$
This formula has a flaw: at $x=1/2$, the right-hand side is strictly zero ($\ln 1$), and we also have the imaginary part evaluated to $-i\pi x$ in the left-hand side, so we should modify the formula, adding more symmetry between $B_-$ and $B_+$:
$$\frac{1}{2} \left(e^{2 i \pi B_- x}+e^{2 i \pi B_+ x}\right)=x \ln \left(\frac{B_+-x}{B_-+x}\right).$$
In this last formula, the left-hand side is also strictly zero at $x=1/2$ because if the argument of an exponent is shifted by $i\pi$, the function changes the sign.
I think, it is a natural relation in umbral algebra. There should be a similar expression with shift of $x$ by $1/2$ but I was not able to find a simplified form for it yet.
Postulating such relation actually maked a huge deal: it connects exponential function with logarithm in closed form. Each can be expressed via the other (for expressing logarithm via exponent one has to make a substitution $\frac{B_+-x}{B_-+x}\to z$ and $x\to\frac{B_+-B_- z}{z+1}$).
This essentially makes all elementary functions expressable via a finite number of additions, multiplications and exponentiations.
So, is it legit to add such an axiom to umbral algebra and what implications would have connecting all elementary functions in closed form?