The generalized Touchard polynomials revisited

We discuss the generalized Touchard polynomials introduced recently by Dattoli et al. as well as their extension to negative order introduced by the authors with operational methods. The connection to generalized Stirling and Bell numbers is elucidated and analogs to Burchnall’s identity are derived. A recursion relation for the generalized Touchard polynomials is established and it is shown that one can interpret some of the resulting formulas as binomial theorems for particular noncommuting variables. We suggest to generalize the generalized Touchard polynomials still further and introduce so called Comtet–Touchard functions which are associated to the powers of an arbitrary derivation.