If we apply operator $D\Delta^{-1}$ to a function, we will get the (Bernoulli) umbral analog of the function. Particularly, applying it to $x^n$ we will get the Bernoulli polynomials $B_n(x)$. Evaluating them at zero we will get the moments of Bernoulli umbra, the Bernoulli numbers $1, -1/2, 1/6,...$ .
But what if we consider the operator $\Delta D^{-1}$? It is the inverse of the aforementioned operator.
What is interesting about it, it corresponds to an umbra with moments $1,1/2,1/3,1/4,...$.
This looks quite trivial, but what are the properties of this "anti-Bernoulli" umbra (let's denote it as $\overline B$)? Denoting the index-lowering operator as $\operatorname{eval}$, the following identities hold:
$\operatorname{eval} \psi(\overline{B}+x)=\ln (x)$
$\operatorname{eval} \ln(\overline{B}+x)=-x \ln (x)+x \ln (x+1)+\ln (x+1)-1$
$\operatorname{eval} \tan(\overline{B}+x)=\frac1\pi\ln \left(\frac{\frac{1}{2}+\frac{x}{\pi }}{\frac{1}{2}-\frac{x}{\pi }}\right)$
Thus, this operator also links exponential functions to logarithms.
Of interest also is the umbra $\overline{B}-1/2$, its moments are reciprocals of https://oeis.org/A001787, with even elements replaced by zeros.
Thus I wonder, what are other properties of umbra $\overline{B}$ and polynomials of it? Are there any known power series expansions with coefficients with denominators from https://oeis.org/A001787 ?
P.S., it seems, the following interesting identity holds:
$\operatorname{eval}\frac{x}{\overline{B} x+1}=\ln (x+1)$
