Sheffer polynomials $\{P_n(x)\}$ have generating function $P(x,t) = \sum_{n=0}^{\infty}P_n(x)t^n=A(t)e^{xu(t)}$.
This form reminds me of the Lie group–Lie algebra correspondence. Is there any connection to Lie theory, obvious or not?
(I've deleted my reference to orthogonal polynomials, which was incorrect and irrelevant here.)
Here are some articles on Sheffer polynomials:
First, a general set of Sheffer polynomials is not orthogonal with respect to some weight function; for example, the prototypical sequence $p_n(x) = x^n$, which belongs to both special sub-groups of Sheffer polynomials—the Appell and binomial Sheffer polynomials—with the e.g.f. $e^{xt}$, is not an orthogonal set—neither is the celebrated Bernoulli Appell Sheffer sequence. (Usually the Sheffer polynomials are defined by $A(t) e^{xB(t)} = \sum_{n\geq 0} p_n(x) \frac{t^n}{n!}= e^{tp_\cdot(x)}$. The normalization $P_n(x)=n!p_n(x)$ often serves to give integer coefficients.)
However, there are connections to constructs associated to Lie theory:
1) The binomial Sheffer sequences have e.g.f.s of the form $e^{xh(t)}$, where $h(0)=0$ and $h'(0) \neq 0$. A dual sequence, its umbral compositional inverse, has the e.g.f. $e^{x\;h^{<-1>}(t)}$ defined by the compositional inverse function. As formulated by Charles Graves as early as 1853, with $g(z) = \frac{1}{\partial_z h(z)}$, the action on a function $f(z)$ analytic at $z$ of the exponential map of the infinitesimal generator (infinigen)
$$ g(z)\frac{\partial }{\partial z} = \frac{1}{h'(z)}\frac{\partial }{\partial z} = \frac{\partial }{\partial h(z)} = \frac{\partial }{\partial \omega} $$
is
(Eqn. 1)
\begin{align} & e^{t\;g(z)\frac{\partial }{\partial z}}\; f(z) = e^{t \;\frac{\partial }{\partial \omega}}\; f(h^{<-1>}(\omega)) \\ = {} & f[h^{<-1>}(\omega + t)] =f[h^{<-1>}(h(z)+t)]. \tag{Eqn. 1} \end{align}
Then
$$e^{t\;g(z)\partial_z}\;z \Big\rvert_{z=0}= h^{<-1>}(t),$$
so the e.g.f. for the associated binomial Sheffer sequence is
$$\left. e^{t\;p.(x)}= e^{x\;h^{<-1>}(t)}= e^{t\;g(z)\partial_z} e^{zx} \;\right\rvert_{z=0}.$$
Edit Oct. 16, 2022: (Start)
Then
$$p_n(x) = (g(z)\partial_z)^n\; e^{zx} \;\Big\rvert_{z=0}$$
$$= e^{-x} (g(y-1)\partial_y)^n e^{yx} \; |_{y=1}= \left. e^{-x} \left[x \;g\left(\frac{u}{x}-1\right)\partial_u\right]^n e^{u} \; \right|_{u=x}.$$
Specializing to $g(z)=(1+z)^{m+1}$, allows a general connection between the normal ordering of iterates of the Scherk-Witt-Lie vectors / infinigens $z^{m+1}\partial_z$ and well-known binomial Sheffer sequences. Then
\begin{align} p_n^{(m)}(x) & =((1+z)^{m+1}\partial_z)^n \;e^{zx}\;\Big|_{z=0} \\[6pt] & = e^{-x}\; (y^{m+1}\partial_y)^n \;e^{yx} \; \Big|_{y=1} \\[6pt] & = e^{-x}\left. \left(x \left(\frac{u}{x} \right)^{m+1} \partial_u\right)^n\; e^u \right|_{u=x} \\[6pt] & = e^{-x}\; x^{-mn} \;(u^{m+1}\partial_u)^n\; e^{u} \; \Big|_{u=x} \\[6pt] & = e^{-x}\; x^{-mn} \;(x^{m+1}\partial_x)\;^n e^x = e^{-x}\; p_n^{(m)}(:x\partial_x:)\; e^x \end{align}
with, by definition, $(:x\partial_x:)^n := x^n\partial_x^n$, implying
(Eqn. 2a)
\begin{align} (x^{m+1} \partial_x)^n & = x^{mn} p_n^{(m)}(:x\partial_x:) \\[6pt] & = x^{mn}\; m^n St1_n^r \left( \frac{x\partial_x}{m} \right) \tag{Eqn. 2a} \end{align}
where the e.g.f. for the $m$-th family of binomial Sheffer polynomials $p_n^{(m)}(x)$ is
(Eqn. 2b)
\begin{align} & e^{tp.^{(m)}(x)} = e^{h^{<-1>}(t)x} \\[6pt] = {} & \exp[((1-mt)^{-\frac{1}{m}}-1)x] \tag{Eqn. 2b} \end{align}
and $St1_n^{r}(x)$ are the reversed unsigned Stirling polynomials of the first kind of OEIS A094638 (see also A008275, A048994, and A130534).
For the special linear Lie algebra $sl_2$:
For $m =-1$, the infingen $g(z)\partial_z = g(1+z)\partial_z= \partial_z$ has associated $h(z)=h^{<-1>}(z)=z$ giving the translation $e^{t\partial_z}\;f(z) = f(z+t)$, consistent with Eqn. (2) with $p^{(-1)}_n(z) = z^n$, the iconic Appell Sheffer sequence, which is also a binomial Sheffer sequence.
For $m=0$, the infingen $g(z)\partial_z = z\partial_z$ has associated $h(z)=\ln(z)$ and $h^{<-1>}(\omega)=e^\omega$ giving, from Eqn. (1), the scaling $e^{tz\partial_z}f(z) = f[\exp(\ln(z)+t)] = f(e^tz)$. Eqn. (2) associated with $g(z)=(1+z)$, $h(z) =\ln(1+z)$, and $h^{<-1>}(\omega) =e^{\omega}-1$ gives
(Eqn. 3)
$$(z\partial_z)^n = p^{(0)}_n(:z\partial_z:) = \operatorname{St2}_n(:z\partial_z:), $$
the Bell / Touchard / Scherk / Stirling polynomials of the second kind with the e.g.f. $\exp[(e^t-1)x]$ (cf. A008277 and A048993). This is easy to corroborate using $(z\partial_z)^n z^k = k^nz^k$.
For $m=1$, the infingen $g(z)\partial_z = z^2\partial_z$ has associated $h(x)=-\frac{1}{z}$ and $h^{<-1>}(\omega)=-\frac{1}{\omega}$ giving, from Eqn. (1), the vertical shearing $e^{tz^2\partial_z}f(z) = f\left [-\frac{1}{-\frac{1}{z}+t}\right] = f(\frac{z}{1-tz})$. Eqn. (2) associated with $g(z)=(1+z)^2$, $h(z) =\frac{z}{1+z}$, and $h^{<-1>}(\omega) =\frac{\omega}{1-\omega}$ gives
$(z^2\partial_z)^n =z^n p^{(1)}_n(:z\partial_z:) = z^n Lah_n(:z\partial_z:)$,
the shifted Lah polynomials, or shifted, normalized, unsigned Laguerre polynomials of order -1, $n! \operatorname{Lag}_n^{<-1>}(-z)$, with the e.g.f. $\exp[z \frac{t}{1-t}]$ (cf. A008297 and A111596). This is corroborated by noting
$$(z^2\partial_z)^n = z\;(z\partial_zz)^n\;z^{-1} = z\;z^n\partial_z^n z^n \;z^{-1} = z^n\; z \;(\partial_z^nz^n)\; z^{-1}$$
$$ = z^n\; n!\; z\; \operatorname{Lag}_n(-:z\partial_z:)\;z^{-1}$$
$$ = z^n\; n!\; \operatorname{Lag}^{(-1)}_n(-:z\partial_z:)= z^n \operatorname{Lah}_n(:z\partial_z:)$$
from the Rodriguez formula for the associated Laguerre polynomial sequences (see this MO-Q for the generalized Laguerre functions, a.k.a., Kummer confluent hypergeometric functions). $\operatorname{Lag}_n(z)$ are the classic Laguerre polynomials (order $0$).
(End)
2) Each Sheffer sequence has a pair of ladder ops—the raising/creation and lowering/destruction/annihilation ops defined by $L \; p_n(x) = n \; p_{n-1}(x)$ and $R \; p_n(x) = p_{n+1}(x)$—satisfying the Graves–Lie bracket of vector fields, the commutator, relation
$$[L,R] = LR-RL = 1,$$
from which the Graves–Pincherle derivative
$$[f(L),R] = f'(L)$$
follows.
There is also the commutator
$$[(g(z)\partial_z)^n,u(z)] = n \; (g(z)\partial_z)^{n-1}$$
for the powers of the Lie derivative / infinitesimal generator for the Scherk-Comtet partition polynomials of A139605, forming an underlying calculus for the binomial Sheffer sequences.
Edit July 1, 2023: (Start)
The typo $u(z)$ just above should be $h(z)$ so that
(Eqn. 4)
$$[(g(z)\partial_z)^n,h(z)] = n \; (g(z)\partial_z)^{n-1}.$$
Proof:
$$e^{tg(z)\partial_z} \;h(z)f(z) - h(z) \;e^{tg(z)\partial_z}\;f(z)$$
$$= (h(z)+t)\; f(h^{<-1>}(h(z)+t)) -h(z)\; f(h^{<-1>}(h(z)+t))$$
$$= t\; f(h^{<-1>}(h(z)+t)) = t e^{tg(z)\partial_z} \;f(z)$$
$$=\sum_{n \geq 0} (g(z)\partial_z)^{n} \frac{t^{n+1}}{n!} \;f(z) =\sum_{n \geq 1}n (g(z)\partial_z)^{n-1} \frac{t^n}{n!}\;f(z) $$
(End)
3) A general Sheffer sequence (a semidirect product of the Appell and binomial Sheffer sequences) is associated with a generalized Lie derivative via
$$e^{t(q(z)+g(z)\partial_z)} \; e^{xz} |_{z=0} = A(t) e^{xf^{<-1>}(t)}$$
with $q(z) = \partial_t \ln(A(t)) \;|_{t=f(z)}$ and $g(z) = 1/f'(z)$.
I have an extensive set of posts at my web blog and links to numerous OEIS entries and MO-Q&As and MSE-Q&As related to this topic. All the symmetric polynomials/functions--complete, elementary, power, Faber--are related to Sheffer Appell sequences with their related ladder ops. The Faa di Bruno/Bell and cycle index polynomials of the symmetric groups, a.k.a the refined Stirling partition polynomials of the second and first kinds, and other compositional partition polynomials are all Appell Sheffer sequences in a distinguished indeterminate. The closely related sets of Lagrange inversion partition polynomials, including the refined Euler characteristic partition polynomials of the associahedra, all have the raising op $g(z)\partial_z$, as shown above--one set, OEIS A134264, is an Appell sequence and is related to free probability, inversion of Laurent series, and characterization of Kac-Schwarz operators related to Heisenberg-Virasoro groups. Multiplicative inversion is intimately bound with the refined Euler characteristic partition polynomials of the permutahedra, or permutohedra (A133314), and the operational (differential/matrix) calculus of Appell Sheffer polynomials. The Bernoulli Appell polynomials are of course related to the BCHD theorem, exponential mappings of the Lie commutator, topology, Todd operator and class of Hirzebruch, the Hurwitz zeta function, and more. Formal group laws (and generalized local Lie groups) are related to the linearization coefficients of products of binomial Sheffer polynomials. The list goes on.
The orthogonal Sheffer sequences such as the associated Laguerre polynomials and families of Hermite polyomials have a lot of underlying group theory and vector fields associated with them. Vilenkin with "Special Functions and the Theory of Group Representations", Talman, Miller, Gilmore, Feinsilver, and many others have written books on the underlying group theory. (None covers every aspect.)
One group of researchers has also written extensively on this topic. See, e.g., "One-parameter groups and combinatorial physics" by Duchamp, Penson, Solomon, Horzela, and Blasiak (although quite negligent of reffing the OEIS--tribal instincts...sigh...What can you do?) as well as Wolfdieter Lang.
4) (Added Oct. 16, 2022) The raising operator of an Appell Sheffer sequence gives the infinigen, related to the Riemann zeta function, for Heaviside fractional differ-integral operators. See, e.g., my MO-Q "Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus" and MO-A to "What's the matrix of logarithm of derivative operator (lnD)? What is the role of this operator in various math fields?".
(Added Aug. 1, 2024) Lie theory, Sheffer hybrid umbral-operational calculus, and Heaviside fractional differ-integral calculus:
Define the Appell reciprocal factorial polynomials $RF_n(x)$ with the e.g.f.
$$ e^{RF.(x)t} = \frac{1}{t!}e^{xt} = e^{tR_x}1$$
where the diff op component of the raising op is given by the digamma function as
$$R_x = \frac{1}{D_x!}xD_x! = \partial_{t = D_x} \ln[e^{RF.(x)t}] = x - \psi(1+D_x),$$
and the lowering op is $L_x = D_x$, as for all Appell sequences. Then
$$ \frac{1}{(\alpha+\beta)!}e^{x(\alpha+\beta)} = e^{ (\alpha+\beta)R_x}1 = e^{\alpha R_x} e^{ \beta R_x}1 = e^{\alpha R_x}\frac{1}{\beta!}e^{x\beta}.$$
Letting $x = \ln(z)$, or $z = e^x$, we have
$L_x =D_x = L_z = zD_z$, the Euler/state-number op, and
$R_x = R_z = \ln(z) - \psi(1+zD_z).$
Then
$$ \frac{z^{\alpha+\beta}}{(\alpha+\beta)!} = e^{ (\alpha+\beta)R_z}1 = e^{\alpha R_z} e^{ \beta R_z}1 = e^{\alpha R_z}\frac{z^{\beta}}{\beta!} = D_z^{-\alpha} \frac{z^{\beta}}{\beta!},$$
identifying
$$D_z^{\alpha} = e^{\alpha \ln(D_z)} = e^{-\alpha R_z}$$
and the infinitesimal generator
$$\ln(D_z) := -R_z = -\ln(z) + \psi(1+zD_z).$$
The Graves-Pincherle derivative transforms to
$$f'(L_x) = [f(L_x),R_x] = [f(L_z),R_z] = [f(zD_z), \ln(z)-\psi(1+zD_z)] = [f(zD_z), \ln(z)],$$
so
$$\partial_{zD_z} f(zD_z) = f'(zD_z) = [-\ln(z),f(zD_z)] = [\ln(D_z),f(zD_z)], $$
consistent with eqn. 4 above with $h(z) = \ln(z)$ and $f(z) = z^n$.
With the Lie bracket and conjugation adjoints for operators defined by $ad_A C =[A,C] = AC-CA$ and $Ad_A C = ACA^{-1},$
the exponential adjoint identity
$$e^{t \; ad_A} = e^{ad_{tA}} = Ad_{e^{tA}}$$
implies
$$e^{t\partial_{zD_z}}f(zD_z) = f(t+zD_z) = e^{-t\;ad_{\ln(z)}} f(zD_z)$$
$$ = e^{t\;ad_{\ln(D_z)}} f(zD_z) = Ad_{e^{t\ln(D_z)}}f(zD_z) = D_z^tf(zD_z)D_z^{-t},$$
(Check consistency with actions of the second and last expressions on $\frac{z^s}{s!}$.)
With $f(zD_z) = \binom{zD_z+\alpha+\beta}{\beta}$, there are connections (see again this MO-Q) to the Kummer confluent hypergeometric functions, ubiquitous in classical and quantum physics as well as number theory, combinatorics, and Lie-Heisenberg-Weyl algebra and group theory (see, e.g., this MO-Q).
The Heaviside infinigen, Newton series, tangent spaces, the Bernoulli function, and a Baker-Campbell-Hausdorff formula:
The Appell Sheffer Bernoulli polynomials $B_n(x)$ and the binomial Sheffer Stirling polynomials of the first and second kinds $ST1_n(x)$ and $ST2_n(x)$ are intertwined (and consequently the Eulerian polynomials, see, e.g., this MO-Q) and manifest in several areas of mathematics, as noted above. The tangent space of the falling factorials $ST1_n(x)$ can be related to the infinigen for the Heaviside differ-integral fractional calculus and to umbral subsitution with the Bernoulli polynomials. (This should be no suprise since the infinigen and the Bernoulli polynomials are intimately related to differentiation.)
The functional compositional inverse (CI) pair $h(x) =e^x-1$ and $h^{<-1>}(x) = \ln(1+x)$ lie at the core of the two sets of Stirling polynomials;
$$e^{ST1.(x)t} = e^{x\ln(1+t)} = (1+t)^x$$
and
$$e^{ST2.(x)t} = e^{x(e^{t}-1)}.$$
This functional CI relation implies the Stirling umbral CI relation
(SUCIR)
$$ST1_n(ST2.(x)) = x^n = ST2_n(ST1.(x))$$
via, e.g.,
(Eqn. 5)
$$ e^{tST2.(ST1.(y))}= e^{ST1.(y)(e^t-1)} = e^{y (\ln(1+e^t-1))} = e^{yt}.$$
The SUCIR also implies the lowering operator for the $ST1_n(x) = n! \binom{x}{n} = x(x-1)...(x-n+1)$ is the finite difference operator $L = \delta_{x} = h(D_x) = e^{D_x}-1$, so
$$ \delta_x \binom{x}{n} =(e^{D_x}-1)\binom{x}{n} = \binom{x+1}{n} -\binom{x}{n}= \binom{x}{n-1}.$$
With $t$ replaced by the diff op $D_x$ in eqn. 5, the SUCIR is revealed to underlie two reps of the iconic Lie group action of translation--the basic shift op embedding $D_x$ and Newtion series embedding $\delta_x$--which, in turn, underlie the finite difference calculus:
$$f(x+y) = e^{yD_x}f(x) = e^{ST1.(y)(e^{D_x}-1)}f(x) = e^{ST1.(y) \delta_{x}}f(x)$$
$$ = \sum_{n \geq 0}\binom{y}{n}\delta_{x}^n f(x) = (1+ \delta_{x})^yf(x)$$
$$ = \sum_{n \geq 0}\binom{y}{n}(e^{D_x}-1))^n f(x) = \sum_{n \geq 0}(-1)^n\binom{y}{n}\sum_{k=0}^n (-1)^k\binom{n}{k}e^{kD_x}f(x)$$
$$ = \sum_{n \geq 0}(-1)^n\binom{y}{n}\sum_{k=0}^n (-1)^k\binom{n}{k}f(x+k).$$
The diff op definition of $ST2_n(x)$ in eqn. 3, $(zD_z)^n = ST2(:zD_z:)$, along with the SUCIR implies
$$\binom{zD_z}{n} = \frac{ST1_n(zD_z)}{n!} = \frac{ST1_n(ST2.(:zD_z:))}{n!} = \frac{:zD_z:^n}{n!}= \frac{z^nD_z^n}{n!}.$$
Then with the Bernoulli numbers $b_n$ and polynomials $B_n(x)$ defined by
$$e^{B.(x)t} = \frac{t}{e^t-1}e^{xt} = e^{b.t}e^{xt} = e^{(b.+x)t} = e^{b.\partial_x}e^{xt} =\frac{D_x}{e^{D_x}-1}e^{xt} $$
and the Appell log polynomials $LP_n(x)$ (see A238363 & this MO-Q) defined by the e.g.f.
$$ e^{LP.(x)t}= \frac{\ln(1+t)}{t}e^{xt} =e^{lp.t}e^{xt}= e^{(lp.+ x)t} =e^{lp.D_x}e^{xt} = \frac{\ln(1+D_x)}{D_x}e^{xt} ,$$
with $lp_n = LP_n(0) = (-1)^n \frac{n!}{1+n}$;
we have the string of identities
$$\left [\ln(D_z),\binom{zD_z}{n+1}\right ] = \partial_{zD_z}\binom{zD_z}{n+1}$$
$$ = \frac{\partial_{zD_z}}{e^{\partial_{zD_z}}-1} (e^{\partial_{zD_z}}-1) \binom{zD_z}{n+1}= e^{b.\partial_{zD_z}} \binom{zD_z}{n} = \binom{B.(zD_z)}{n}$$
$$ = \frac{\ln(1+\delta_{zD_z})}{\delta_{zD_z}} \binom{zD_z}{n} = \sum_{k \geq 0} \frac{(-1)^k}{k+1} \delta_{zD_z}^k \binom{zD_z}{n} = \sum_{k = 0}^n \frac{(-1)^k}{k+1} \binom{zD_z}{n-k}$$
$$ =\frac{1}{n!} \sum_{k = 0}^n\binom{n}{k} \frac{(-1)^k k!}{k+1} ST1_{n-k}(zD_z) = \frac{1}{n!} \sum_{k = 0}^n \binom{n}{k} \frac{(-1)^k k!}{k+1} z^{n-k}D_z^{n-k}= \frac{LP_n(:zD_z:)}{n!} $$
$$ =\left [\binom{zD_z}{n+1}, \ln(z)\right ] = \left [\frac{:zD_z:^{n+1}}{(n+1)!}, \ln(z)\right ].$$
This implies the umbral conjugation adjoint relations
$$LP_n(x) = ST1_n(B.(ST2(x))) =: \widetilde{Ad}_{ST1_n} B.$$
and
$$B_n(x)= ST2_n(LP.(ST1.(x))) =:\widetilde{Ad}_{ST2_n} LP. ,$$
or, in terms of lower triangular coefficient matrices,
$$[LP] = [ST1][B][ST2] = [ST1][B][ST1]^{-1} = Ad_{[ST1]}[B]$$
and
$$[B] = [ST2][LP][ST1] = [ST2][LP][ST2]^{-1} = Ad_{[ST2]}[LP].$$
Note also
$$D_x^m \binom{x}{n} = [\frac{D_x}{e^{D_x}-1}(e^{D_x}-1)]^m\binom{x}{n} = \langle e^{b.D_x}\rangle^m\delta_x^m \binom{x}{n} = \langle e^{b.D_x}\rangle^m \binom{x}{n-m}= \binom{B.^{(m-1)}(x)}{n-m},$$
identifying the Appell Norlund convolutional m-th order Bernoulli polynomial sequences with the e.g.f.s
$$ e^{B.^{(m-1)}(x)t} = \left(\frac{t}{e^t-1}\right)^m e^{xt} = \langle e^{b.t}\rangle^m e^{xt}$$
with $\langle ...\rangle$ denoting umbral evaluation before the exponentiation (for some more umbral relations, see my blog post "The Hirzebruch criterion for the Todd class").
Returning to the Newton series, a more robust expression for the Bernoulli/Todd op and an extension of the Bernoulli polynomials to the Bernoulli function can be constructed in which the Bernoulli function lives in the tangent space (w.r.t. $x$) of the Hurwitz zeta function.
First, with
$$\delta_xf(x) = (e^{D_x}-1)f(x) = f(x+1)-f(x) = f_{-}(x) ,$$
the derivative of a function can be expressed in terms of its finite differences as
$$D_xf(x+y) =e^{yD_x}D_x f(x) $$
$$= e^{ST1.(y)\delta_x}D_x f(x) = (1+ \delta_{x})^y\frac{\ln(1+ \delta_{x})}{\delta_{x}}(\delta_xf(x))$$
$$= D_yf(x+y) = \sum_{n \geq 0}\binom{B.(y)}{n}\delta_{x}^n (\delta_xf(x))$$
$$ = \sum_{n \geq 0}(-1)^n\binom{B.(y)}{n}\sum_{k=0}^n (-1)^k\binom{n}{k} (\delta_xf(x+k))$$
$$ = f_{-}(x+B.(y)) = f_{-}(B.(x+y)) = f(B.(x+y+1)) - f(B.(x+y))$$
$$ = e^{B.(y)D_x}f_{-}(x)= e^{yD_x}\frac{D_x}{e^{D_x}-1}f_{-}(x) ,$$
where the equality is understood as equivalence when convergence holds and under analytic continuation or interpolation, e.g., via a Mellin transform or equivalent Hankel-contour complex integral.
Nota bene $f'(x+y) = f(B.(x+y+1)) - f(B.(x+y))$ is the operational definition of the Bernoulli polynomials, in which the Bernoulli polynomials morph a finite difference into a derivative. (This is at the core of the Bernoulli/Todd operator discretizing integrals and discrete measurements of the volumes of convex polytopes.)
With $y=0$ and $f(x) = \zeta(s,x)$, the Hurwitz zeta function, then $\delta_xf(x) = f_{-}(x) =- x^{-s}$ and we obtain the Bernoulli function $B_s(x)$ as a variant of the Helmut Hasse formula for the Hurwitz zeta;
$$-\partial_x \zeta(s,x) = s\zeta(s+1,x) = \sum_{n \geq 0}(-1)^n\binom{B.(0)}{n}\sum_{k=0}^n (-1)^k\binom{n}{k} (x+k)^{-s}$$
$$= \sum_{n \geq 0}(-1)^n\binom{b.}{n}\sum_{k=0}^n (-1)^k\binom{n}{k} (x+k)^{-s}$$
$$=\sum_{n \geq 0} \frac{1}{n+1}\sum_{k=0}^n (-1)^k\binom{n}{k} (x+k)^{-s}$$
$$=\frac{\ln(1+ \delta_{x})}{\delta_{x}} x^{-s} = e^{lp.\; \delta_x} x^{-s}$$
$$= \frac{D_x}{e^{D_x}-1} x^{-s} = e^{b.D_x} x^{-s} = (b.+x)^{-s} = B.(x)^{-s} = B_{-s}(x)$$
$$ =\int_{0}^{\infty} e^{-(b.+x)t}\ \frac{t^{s-1}}{\left(s-1\right)!}dt =\int_{0}^{\infty}\frac{-t}{e^{-t}-1}\ e^{-xt}\ \frac{t^{s-1}}{\left(s-1\right)!}dt ,$$
(The last Mellin interpolation integral can be morphed into a Hankel contour integral with a greater domain of convergence. Again equality is understood to be a definition when one actual operation leads to a divergent/asymptotic result and another a convergent one.)
The UCI function for the Bernoulli function is the sliding average
$$\hat{B}_s(x) = \frac{(x+1)^{s+1}-x^{s+1}}{s+1} = \int_0^1 (x+t)^sdt = \int_x^{x+1} t^s dt$$
$$ = \frac{e^{D_x}-1}{D_x} x^s = e^{\hat{b}.D_x}x^s = (\hat{b}.+x)^s = \frac{\delta_x}{\ln(1+\delta_x)}x^s = e^{\hat{lp}.\delta_x}x^s$$
$$= \int_{0}^{\infty} e^{-(\hat{b}.+x)t}\ \frac{t^{-s-1}}{\left(-s-1\right)!}dt =\int_{0}^{\infty}\frac{e^{-t}-1}{-t}\ e^{-xt}\ \frac{t^{-s-1}}{\left(-s-1\right)!}dt ,$$
with $\hat{b}_n = \frac{1}{n+1}$, the reciprocal integers, and $\hat{lp}_n$ are the Cauchy numbers. In the limit, $B_{-1}(x) = \ln(1+x)-\ln(x) = \ln\left(\frac{1+x}{x}\right)$.
The operators and identities above are sufficient to derive standard BCH theorems found in treatises on Lie theory.
With $g(z) = \frac{1}{f'(z)}$, we have the special disentangling/BCH duality
$$ e^{t w(z)}e^{tg(z)\partial_z} = e^{t q(z) + t g(z)\partial_z} $$
with, formally,
$$q(z) = w(f^{<-1>}(tB.(f(z)/t)))= w[FL(z, t\cdot b.)] =: e^{t\cdot b.g(z)\partial_z} w(z)$$
and
$$w(z) = q(f^{<-1>}(t\hat{B}.(f(z)/t))) = q[FL(z, t\cdot \hat{b}.)] =: e^{t\cdot \hat{b}.g(z)\partial_z} q(z) $$
with
$$e^{tg(z)\partial_z}z = f^{<-1>}(f(z)+t) = FL(z,t).$$
Check with $q(z) = f(z).$ Then $[g(z)\partial_z,f(z)] = g(z)f'(z)=1$ and $w(z) = q(f^{<-1>}(t\hat{B}.(f(z)/t))) = t \hat{B}_1(f(z)/t)= f(z) + t/2 ,$ so
$$e^{t(f(z) + g(z)\partial_z)} = e^{t^2/2} e^{tf(z)}e^{tg(z)\partial_z}.$$
Conversely, with $w(z) = f(z)$, then $q(z) = w(f^{<-1>}(tB.(f(z)/t))) = t B_1(f(z)/t)= f(z) - t/2$, so
$$e^{t(-t/2+ f(z) + g(z)\partial_z)} = e^{tf(z)}e^{tg(z)\partial_z}.$$
With $f(x) = \frac{x^{r+1}}{r+1}$ where $r$ is any real number (take limits for $r=-1$), then $f^{<-1>}=((r+1)x)^{\frac{1}{r+1}}$, $FL(x,r) = (z^{r+1} +(r+1)t)^{\frac{1}{r+1}}$, $g(x)D_x =x^{-r}D_x$, and, with $q(x) = x^s$,
$$w(x) = q[f^{<-1>}(t\hat{B}.(f(x)/t)] $$
$$= [(r+1)t\hat{B}.\left (\frac{x^{r+1}}{(r+1)t} \right )]^{\frac{s}{r+1}} $$
$$= ((r+1)t)^{\frac{s}{r+1}}\hat{B}_{\frac{s}{r+1}}\left (\frac{x^{r+1}}{(r+1)t} \right )$$
$$ =\ \left(\left(r+1\right)\ t\right)^{\left(\frac{s}{r+1}\right)}\ \frac{\left(1+\frac{x^{\left(r+1\right)}}{\left(r+1\right)t}\right)^{\left(\frac{s}{r+1}+1\right)}-\left(\frac{x^{\left(r+1\right)}}{\left(r+1\right)t}\right)^{\left(\frac{s}{r+1}+1\right)}}{\frac{s}{r+1}+1}$$
$$=\ \frac{x^{\left(r+s+1\right)}}{r+s+1}\ \frac{1}{t}\left(\left(1+t\left(r+1\right)x^{\left(-r-1\right)}\right)^{\frac{\left(s+r+1\right)}{r+1}}-1\right).$$
The last two expressions are formatted for copying & pasting into the online Desmos graphing app for real $r,s,t,$ and $x$ for comparison of domains of agreement and continuity through $x = 0$. The last expression is in agreement with Eqn. $I.2.34$ on pg. 8 of "Evolution operator equations: Integration with algebraic and finite difference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory" by Dattoli, Ottavani, and Torre with $A = tx^s$ and $B = tx^{-r}D_x$.
For $s \to 1$ and $r\to -1$,
$$\frac{z^{r+s+1}}{r+s+1}\frac{\left(1+(r+1)tz^{-r-1} \right)^{\frac{r+s+1}{r+1}}-1}{t} \to z \frac{e^t-1}{t}$$
which agrees with Eqn. $I.2.30$ on pg. 7 of Dattoli et al..
