Goin’ with the Flow: Logarithm of the Derivative Operator is a pdf set of notes under construction on the relations between the commutator of the logarithm of the derivative operator, the Pincherle derivative, Lie operator derivatives, and the two umbrally inverse pair of Sheffer sequences the Bell polynomials and falling factorials. The action of the commutator / Lie op derivative is examined on fractional derivatives of the Riemann-Liouville type as represented by the Kummer confluent hypergeometric functions (generalized Laguerre functions). Matrix reps are presented that illustrate the matrix rep of the Lie operator derivative as a conjugation of the matrix representing the usual derivative op by the Stirling number matrices (first and second kinds). Relations to the n-simplices are also given.
Erratum (Aug. 18, 2015): The Kummer confluent hypergeometric functions on pages 3 and 4 should be multiplied by to give the correct equalities.
The reference MO-Q107159 on page 3 is to the MathOverflow question Pochhammer symbol of a differential and hypergeometric polynomials .