Higher order derivatives of exponential functions and generalized forms of Kampé de Fériet-Bell polynomials (Q2739237)

The Kampé de Fériet (KdF) polynomials are generalizations of the well-known Hermite polynomials. In this paper an extension of the KdF-polynomials, the so-called Kampé de Fériet-Bell (KdF-B) polynomials, defined by means of a generating function of the exponential type \(e^{f(x)}\) with \(f(x)= \sum^\infty_{s=1} a_sx^s\), is treated. Some properties and in particular, relations with Bessel type functions of many variables, are shown. The KdF-B polynomials are also generalized to the multi-index case through a certain generating function in two variables, being those polynomials specially used for the expansion of multivariable exponential functions. Some examples are also given.