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$$ (\mathrm{e}^{b D} f)(x) = f(x + b) $$
$$ (a^{x D} f)(x) = f(x a) $$
Is there a similar expression for the following?
$$ \left(\frac{1}{1-D} f\right)(x) = \left(\sum_{n=0}^\infty D^n f\right) (x) $$
What are the properties of this operator?
Edit: Solving
\begin{align} (1 - D)^{-1} f &= g \\ f &= (1 - D) g \\ &= g - D g \end{align}
for $g$ yields
$$ g(x) = a \mathrm{e}^x - \mathrm{e}^x \int_b^x \mathrm{e}^{-t} f(t) \,\mathrm{d}t $$
as can be shown by taking the derivative of this expression.
Indeed, your formal (edit) expression $$ (1-D)^{-1} f (x)= e^x \int_x^b dt~ e^{-t} f(t) ~, $$ is standard in the seat-of-the-pants bag of tricks utilized in physics, associated with the name of Schwinger, and it is but the obverse of the very Lagrange translation operator, your first formula. I'm skipping your leading term, as it merely underscores the arbitrariness of the limit b, the kernel (homogeneous solution) of 1–D. That's why I'll be cavalier with the lower limit below--adjust as desired with apologies and a wink to mathematicians.
The standard Schwinger trick here amounts to the formal wisecrack, $$ \frac{1}{1-D} = \int_{-\infty}^{0} dt ~ e^{t(1-D)} $$ with the lower limit to be adjusted in comportance with circumstantial exigencies. Thus, from your Lagrange translation operator, $$ \frac{1}{1-D} ~f(x)= \int_{-\infty}^{0} dt ~ e^{t} ~f(x-t) =\int^{\infty}_{x} dt ~ e^{x-t} ~f(t), $$ the arbitrary upper limit adjusted to an arbitrary constant as discussed.
To test-drive it, apply to $f(x)=\exp(-x/2)$, so that $$ \frac{1}{1-D} ~ e^{-x/2}= e^x \left( a+\frac {2}{3} e^{-3x/2}\right ) = \frac{2}{3}e^{-x/2}, $$ where the constant a was chosen to vanish by the evident requirement of vanishing at infinite x.
