The generalized Touchard polynomials revisited (Q2016353)

Starting with the Touchard polynomials, defined by \[ T_n(x)=e^{-x}\left( x \frac{d}{dx}\right) ^ne^x \] for every \(n\in \mathbb N\), several generalizations were recently defined, e.g., for higher order \[ T_n^{(m)}(x)=e^{-x}\left( x^m\frac{d}{dx}\right) ^ne^x \] for positive but also negative integers \(m\). The paper deals extensively with the case \(m=-1\) and among other things suggests to consider generalized Touchard functions of arbitrary real order. As a particular example the case of order \(m=\frac{1}{2}\) is discussed and it is shown that the associated Touchard functions are given by Hermite polynomials. Further generalizations are given (e.g., so called Comtet-Touchard functions associated to powers of an arbitrary derivation are introduced).