In umbral notation and maneuvers, it's quite simple. With an obvious change of notation, let the Stirling polynomials of the first kind (a binomial Sheffer sequence of polynomials in the hybrid umbral-finite operator calculus) be denoted by

$(ST1.(x))^n = ST1_n(x) = \sum_{k=0}^n ST1_{n,k}x^k$

and similarly for the binomial Sheffer polynomials $ST2_n(x).$

The e.g.f.s are

$$e^{x\ln(t+1)} = e^{ST1.(x)t}$$

and

$$e^{x(e^t-1)} = e^{ST2.(x)t}.$$

Then umbral composition gives

$$e^{ST2.(ST1.(x))t} = e^{ST1.(x)(e^t-1)} = e^{x\ln[1+(e^t-1)]} = e^{xt},$$

and, of course,

$$e^{ST1.(ST2.(x))t} = e^{ST2.(x)(\ln(1+t))} = e^{x[e^{\ln(1+t)}-1]} = e^{xt},$$

implying that the two polynomial sequences form an umbral compositional inverse pair, i.e.,

$$ST2_n(ST1.(x))= x^n = ST1_n(ST2.(x))$$

In terms of the pair of lower triangular matrices of coefficients of the two sequences, this implies the pair of matrices is an inverse pair.

The diff op reps of the two binomial Sheffer sequences are

$$ST2_n(:xD:) = (xD)^n$$

and

$$ST1_n(xD) = :xD:^n = x^nD^n,$$

where $:xD:^k = x^kD^k$ by definition with $D = \partial_x$. Then we have the diff op equivalent of the umbrally compositional inverse relation:

$$ST2_n(:xD:) = ST2_n(ST1.(xD)) = (xD)^n$$

and

$$ST1_n(xD) = ST1_n(ST2.(:xD:)) = :xD:^n.$$

The e.g.f.s follow from the conjugation $Ad_{e^{-x}}$ as

$$e^{ST2.(x)t} = e^{-x} e^{ST2(:xD:)t}e^x = e^{-x} e^{txD}e^x =e^{-x}\sum_{n\geq 0} e^{tn}\frac{x^n}{n!} = e^{x(e^t-1)}$$

and from $Ad_{x^{-y}}$ as

$$e^{ST1.(y)t} = x^{-y} e^{ST1.(xD)t}x^y = x^{-y} e^{t:xD:}x^y =\sum_{n\geq 0} \binom{y}{n}t^n = (1+t)^y = e^{y\ln(1+t)}.$$

In addition, the lowering ops defined for any Sheffer polynomial sequence $P_n(x)$ by

$$L_P P_n(x) = n \; P_{n-1}(x)$$

are given by the inverse functions in the arguments of the e.g.f.s as

$$L_{ST1} = e^{D}-1$$

and

$$L_{ST2} = \ln(1+D).$$

If we go one step further and define the finite difference operator

$$\delta_x = e^{\partial_x}-1 = L_{ST1},$$

then

$$\partial_x = \ln(1 + \delta_x),$$

and we have an entre into the finite difference calculus with reps of the (Heaviside fractional) differ-integral calculus and the Appell Sheffer Bernoull polynomials and function (essentially the Hurwitz zeta function) along with their umbral inverses with their connections to the Bernoulli-Todd-Hirzebruch operator and class, exact Euler-MacLaurin series, volumes of lattice polytopes, the BCH theorem, and more via the ops

$$ \frac{\partial_x}{\delta_x} = \frac{\partial_x}{e^{\partial_x}-1} =
\frac{\ln(1+\delta_x)}{\delta_x}$$

and their multiplicative inverse ops

$$ \frac{\delta_x}{\partial_x} = \frac{e^{\partial_x}-1}{\partial_x} = \frac{\delta_x}{\ln(1+\delta_x)}.$$

signedStirling numbers of the 1st kind, whereas $c(n,k)=|s(n,k)|$ is used for their absolute values. For instance, this is what Wikipedia uses: en.wikipedia.org/wiki/Stirling_number $\endgroup$7more comments