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Added 4/8/2024:

The arguments above are consistent With the presentation of $\ln(D)$ in The Geometry of Infinite-Dimensional Groups by Boris Khesin and Robert Wendt. On pg. 113, the authors state

Below we are going to define the logarithm of the derivative operator $\partial$. ... this will be an operator not on functions, but on pseudodifferential symbols, and it appears rather useful in describing their central extensions.

Applications are presented in the article. Their presentation essentially relies on a commutator based on a generalized Leibniz formula, asserted by the authors but which has been proven by Osler (see "Fractional generalizations of Leibniz' formula" by N. Wheeler) for the FID op I present here. With $D_L$ and $D_R$ denoting action of $D_x$ on the left and right side, respectively, of a product of functions,

$$D_x^{\alpha}f(x)g(x) = (D_R + D_L)^{\alpha} f(x)g(x)$$ $$ = \sum_{k \geq 0} \binom{\alpha}{k} D_L^k D_R^{\alpha-k}f(x)g(x)$$

$$ = \sum_{k \geq 0} \binom{\alpha}{k} (D_L^k f(x)) (D_R^{\alpha-k}g(x)) = \sum_{k \geq 0} \binom{\alpha}{k} (D_x^k f(x)) (D_x^{\alpha-k}g(x)) ,$$

and the associated commutator is (Eqn. 4.34 in K&W)

$$[D^{\alpha}_x , f(x)D^n_x] = \sum_{k \geq 1} \binom{\alpha}{k} (D_x^k f(x)) D_x^{\alpha+n-k} .$$

Then evaluating this with $D_{\alpha=0}$ gives the commutator (Eqn. 4.36)

$$[\ln(D_x), f(x)D_x^n] = \sum_{k \geq 1} \frac{(-1)^{k+1}}{k} (D_x^k f(x)) D_x^{n-k} .$$

With $f(x) = x$ and $n=0$, this gives, in agreement with the Pincherle Derivative formulation above,

$$[\ln(D_x), x] = D_x^{-1}.$$

K & W propose

Definition / Proposition 4.8 The linear map $[\log(\partial),\;.\;] : \Psi DS \to \Psi DS$ given by formula (4.36) is an outer derivation of the Lie algebra $\Psi DS $ of pseudodifferential symbols.

$$D_x^{-1} \frac{x^{\omega}}{\omega!} = \frac{x^{\omega+1}}{(\omega+1)!}.$$

$$ \frac{x^{\alpha+\omega}}{(\alpha+\omega)!} = \bigtriangledown^{\alpha}_{n}\bigtriangledown^{n}_{k}\frac{x^{\omega+k}}{(\omega+k)!}$$

$$ = \bigtriangledown^{\alpha}_{n}\bigtriangledown^{n}_{k} D_x^{-k} \;\frac{x^{\omega}}{\omega!}$$

$$ = [1-(1-D_x^{-1})]^{\alpha} \; \;\frac{x^{\omega}}{\omega!} = D_x^{-\alpha}\;\frac{x^{\omega}}{\omega!}. $$

For $\alpha = -m$ with $m = 1,2,...$ and $\omega = 0$, this Newton interpolator gives

$$\frac{1}{1-c.t} = \int_0^{\infty} \frac{e^{-\frac{1}{1-t}x}}{1-t} \frac{x^{\alpha}}{\alpha!} \; dx = (1-t)^{\alpha},$$

for $Re(\alpha) > -1$, giving

$$D_x^{-1} \frac{x^\omega}{\omega!} = \frac{x^{\omega+1}}{(\omega+1)!}.$$

$$ \frac{x^{\alpha+\omega}}{(\alpha+\omega)!} = \bigtriangledown^{\alpha}_n \bigtriangledown^n_k \frac{x^{\omega+k}}{(\omega+k)!}$$

$$ = \bigtriangledown^\alpha_n \bigtriangledown^n_k D_x^{-k} \;\frac{x^\omega}{\omega!}$$

$$ = [1-(1-D_x^{-1})]^\alpha \; \;\frac{x^\omega}{\omega!} = D_x^{-\alpha}\;\frac{x^\omega}{\omega!}. $$

For $\alpha = -m$ with $m = 1,2,\ldots$ and $\omega = 0$, this Newton interpolator gives

$$\frac{1}{1-c.t} = \int_0^\infty \frac{e^{-\frac{1}{1-t}x}}{1-t} \frac{x^\alpha}{\alpha!} \; dx = (1-t)^\alpha,$$

for $\operatorname{Re}(\alpha) > -1$, giving

$$D_x^{\alpha} \; H(x) \; \frac{x^{\omega}}{\omega!} = H(x) \frac{x^{\omega-\alpha}}{(\omega-\alpha)!} ,$$

with $n=0,1,2,3,...$.

Note this has little to do with a Fourier transform over the real line or any pseudo-diff op/symbol associated with such. In particular, $D^{\alpha}$ here is NOT associated with multiplication by $(i 2 \pi f)^{\alpha}$ in frequency space. Elsewhere I show various equivalent convolutional reps of this FID as 1) a FT over a circle via a transformation of a regularized Cauchy complex contour integral, 2) the analytic continuation of the integral rep of the Euler beta function either through a blow-up into the complex plane of the integral along the real line segment or regularization via the Hadamard finite part or via the Pochhammer contour, 3) the Mellin interpolation of the standard derivative operator via the action of the generating function $e^{tD_x}$, an operator application of Ramanujan's master formula, or 4) a sinc function/cardinal series interpolation of the generalized binomial coefficients.

$$ e^{\alpha \; IG} \; H(x) \; \frac{x^{\omega}}{\omega!} = D_x^{\alpha} \; H(x) \; \frac{x^{\omega}}{\omega!} = H(x) \frac{x^{\omega-\alpha}}{(\omega-\alpha)!} = e^{-\alpha D_{\omega}} \; H(x) \; \frac{x^{\omega}}{\omega!},$$

$$D_{\alpha} \; e^{\alpha IG} \; H(x) \; \frac{x^{\omega}}{\omega!} |_{\alpha =0} = IG \; H(x) \; \frac{x^{\omega}}{\omega!} = \ln(D_x) \; H(x) \; \frac{x^{\omega}}{\omega!}$$

$$ = D_{\alpha} \; H(x) \; \frac{x^{\omega-\alpha}}{(\omega-\alpha)!} |_{\alpha =0} = -D_{\omega} \;\frac{x^{\omega}}{\omega!}$$

$$ = [ \; -\ln(x) + \psi(1+xD_x) \;] \; H(x) \; \frac{x^{\omega}}{\omega!}, $$

where $\psi(x)$ is the digamma function, which can be defined over the complex plane as a meromorphic function and is intimately related to the values of the Riemann zeta function at $s = 2,3,4,...$.

$$IG \; f(x)=\frac{1}{2\pi i}\oint_{|z-x|=|x|}\frac{-\ln(z-x)+\lambda}{z-x}f(z) \; dz$$

$$=(-\ln(x)+\lambda) \; f(x)+ \int_{0}^{x}\frac{f\left ( x\right )-f(u)}{x-u}du$$

$$ = [\; -\ln(x)+ \frac{\mathrm{d} }{\mathrm{d} \beta}\ln[\beta!]\mid _{\beta =xD} \; ] \; f(x)=[ \; -\ln(x)+\Psi(1+xD) \;] \; f(x)$$

$$ = [ \; -\ln(x)+\lambda - \sum_{n=1}^{\infty } (-1)^n\zeta (n+1) \; (xD)^n \;] \; f(x)$$

where $\lambda$ is related to the Euler-Mascheroni constant via $\lambda=D_{\beta} \; \beta! \;|_{\beta=0}$.

$$R_z = \frac{1}{D_z!} \; z \; D_z! = z + [\frac{1}{D_z!},z] \; D_z! .$$

Now re-enter Pincherle and the eponymous operator derivative, which Rota touted for the finite operator calculus. The Graves-Pincherle derivative derives its power from the Graves-Lie-Heisenberg-Weyl commutator $[D_z,z] = 1$ from which, by normal re-ordering, implies for any function expressed as a power series in $D_z$

$$[f(D_z),z] = f'(D_z) = D_t \; f(t) \; |_{t = D_z}.$$

This is an avatar of the Pincherle derivative (PD) that follows from the action $$[D^n,z] \; \frac{z^{\omega}}{\omega!} = [\;\frac{\omega+1}{(\omega+1-n)!} - \frac{1}{(\omega-n)!}\;] \; z^{\omega+1-n} = n \; D_z^{n-1} \; \frac{z^{\omega}}{\omega!},$$

$$R_z = \frac{1}{D_z!} \; z \; D_z! = z + [\frac{1}{D_z!},z] \; D_z! = z + D_{t = D_z}\; \ln[\frac{1}{t!}] $$

$$D_x^\alpha \; H(x) \; \frac{x^\omega}{\omega!} = H(x) \frac{x^{\omega-\alpha}}{(\omega-\alpha)!} ,$$

with $n=0,1,2,3,\ldots$.

Note this has little to do with a Fourier transform over the real line or any pseudo-diff op/symbol associated with such. In particular, $D^\alpha$ here is NOT associated with multiplication by $(i 2 \pi f)^\alpha$ in frequency space. Elsewhere I show various equivalent convolutional reps of this FID as 1) a FT over a circle via a transformation of a regularized Cauchy complex contour integral, 2) the analytic continuation of the integral rep of the Euler beta function either through a blow-up into the complex plane of the integral along the real line segment or regularization via the Hadamard finite part or via the Pochhammer contour, 3) the Mellin interpolation of the standard derivative operator via the action of the generating function $e^{tD_x}$, an operator application of Ramanujan's master formula, or 4) a sinc function/cardinal series interpolation of the generalized binomial coefficients.

$$ e^{\alpha \; IG} \; H(x) \; \frac{x^\omega}{\omega!} = D_x^{\alpha} \; H(x) \; \frac{x^\omega}{\omega!} = H(x) \frac{x^{\omega-\alpha}}{(\omega-\alpha)!} = e^{-\alpha D_\omega} \; H(x) \; \frac{x^\omega}{\omega!},$$

$$D_\alpha \; e^{\alpha IG} \; H(x) \; \left. \frac{x^\omega}{\omega!} \right|_{\alpha =0} = IG \; H(x) \; \frac{x^\omega}{\omega!} = \ln(D_x) \; H(x) \; \frac{x^\omega}{\omega!}$$

$$ = D_{\alpha} \; H(x) \; \left. \frac{x^{\omega-\alpha}}{(\omega-\alpha)!} \right|_{\alpha =0} = -D_{\omega} \;\frac{x^{\omega}}{\omega!}$$

$$ = [ \; -\ln(x) + \psi(1+xD_x) \;] \; H(x) \; \frac{x^\omega}{\omega!}, $$

where $\psi(x)$ is the digamma function, which can be defined over the complex plane as a meromorphic function and is intimately related to the values of the Riemann zeta function at $s = 2,3,4,\ldots$.

$$IG \; f(x)=\frac{1}{2\pi i} \oint_{|z-x|=|x|}\frac{-\ln(z-x)+\lambda}{z-x}f(z) \; dz$$

$$=(-\ln(x)+\lambda) \; f(x)+ \int_0^x \frac{f(x)-f(u)}{x-u}\, du$$

$$ = \left[\; -\ln(x)+ \left. \frac{\mathrm{d} }{\mathrm{d} \beta} \ln[\beta!]\right| _{\beta =xD} \; \right] \; f(x)= \left[ \; -\ln(x)+\Psi(1+xD) \;\right] \; f(x)$$

$$ = \left[ \; -\ln(x)+\lambda - \sum_{n=1}^\infty (-1)^n\zeta (n+1) \; (xD)^n \;\right] \; f(x)$$

where $\lambda$ is related to the Euler–Mascheroni constant via $\lambda=D_\beta \; \beta! \;|_{\beta=0}$.

$$R_z = \frac{1}{D_z!} \; z \; D_z! = z + \left[\frac{1}{D_z!},z \right] \; D_z! .$$

Now re-enter Pincherle and the eponymous operator derivative, which Rota touted for the finite operator calculus. The Graves–Pincherle derivative derives its power from the Graves-Lie-Heisenberg-Weyl commutator $[D_z,z] = 1$ from which, by normal re-ordering, implies for any function expressed as a power series in $D_z$

$$[f(D_z),z] = f'(D_z) = D_t \; f(t) \; \Big|_{t = D_z}.$$

This is an avatar of the Pincherle derivative (PD) that follows from the action $$[D^n,z] \; \frac{z^\omega}{\omega!} = \left[\;\frac{\omega+1}{(\omega+1-n)!} - \frac{1}{(\omega-n)!}\;\right] \; z^{\omega+1-n} = n \; D_z^{n-1} \; \frac{z^\omega}{\omega!},$$

$$R_z = \frac{1}{D_z!} \; z \; D_z! = z + \left[\frac{1}{D_z!},z \right] \; D_z! = z + D_{t = D_z}\; \ln\left[\frac{1}{t!}\right] $$