Generalized Bell polynomials

In this paper, generalized Bell polynomials ( b phi n ) n associated to a sequence of real numbers phi = ( phi i ) infinity i =1 are introduced. Bell polynomials correspond to phi i = 0, i >= 1. We prove that when phi i >= 0, i >= 1: (a) the zeros of the generalized Bell polynomial b phi n are simple, real and non positive; (b) the zeros of b phi n +1 interlace the zeros of b phi n ; (c) the zeros are decreasing functions of the parameters phi i . We find a hypergeometric representation for the generalized Bell polynomials. As a consequence, it is proved that the class of all generalized Bell polynomials is actually the same class as that of all Laguerre multiple polynomials of the first kind. (c) 2024 Published by Elsevier Inc.