Can we add additional structure to the umbral ring?

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

$$\frac{1}{2} \left(e^{2 i \pi B_- x}+e^{2 i \pi B_+ x}\right)$$ and $$x \ln \left(\frac{B_+-x}{B_-+x}\right)$$

where $B_-$ is Bernoulli umbra (an umbra with moments equal to Bernoulli numbers) and $B_+=B_-+1$.

They both have the evalutaion (scalar part) equal to

$$\pi x \cot (\pi x)=1-\frac{\pi ^2 x^2}{3}-\frac{\pi ^4 x^4}{45}-\frac{2 \pi ^6 x^6}{945}-\frac{\pi ^8 x^8}{4725}-\frac{2 \pi ^{10} x^{10}}{93555}-\dots$$

The question is, can we somehow link them, adding more structure to the umbral ring?

$$\frac{1}{2} \left(e^{2 i \pi B_- x}+e^{2 i \pi B_+ x}\right)\stackrel{?}{=}x \ln \left(\frac{B_+-x}{B_-+x}\right)$$

It seems, this cannot be the equality everywhere. For instance, the left-hand part has zeroes at $x=1/2+kx$, $k\in\mathbb{Z}$, while the right-hand part cannot have there zeroes as it would mean that logarithm of something other than $1$ is zero.

But we can speculate that the equality can hold on $x\in(0,1)$ (at $x=0$ and $x=1$ the right-hand side is undefined because $\ln B_-$ has the scalar part diverging). Will this rule break something or add a rich structure to the ring? Maybe the equality can be modified to make it more universal?

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