Left and right acting derivatives

Saw an interesting notation in this Math Stackexchange answer. I quote -

“

Eq. may be rewritten as , where acts on the left and on the right, hence

by the binomial theorem which can be applied straightforwardly, since and commute. Now, this result can be extended to non-integer by using [generalized binomial coefficients][1], leading to the following von Neumann series :

where the equivalent symbol means that the this relation has to be considered when applied to products of functions. Eq. corresponds with the case .

As a final note, let’s remark that the geometric expansion is recovered due to the fact that the exponential function is an eigenvector of the derivative operator, whence

”