The reviewer [Nederl. Akad. Wet., Proc., Ser. A 79, 457-461 (1976; Zbl 0335.33002)] considered a class of generalized Hermite polynomials generated by \(G[(p+1)xt- t^{p+1})]\), where \(p\) is a positive integer and \(G[z]\) is assumed to possess an analytic expansion about \(z=0\) with nonzero coefficients. For members of this general class of polynomials with \(G[z]= \exp(z)\), the author derives several interesting properties and characteristics by using certain operational rules associated with the monomiality principle.