An Extension of a Class of Polynomials. II

In this paper we study the algebraic structure of the class of polynomials $\{ u_n !H_n (x)\} $ in x, where $\{ {H_n (x)} \}$ satisfies the functional equation $D_u H_n (x) = H_{n - 1} (x)$ for $n = 1,2, \cdots $, and where $D_u $ is a general operator, linear and distributive, which transforms a polynomial of degree n in x into one of degree $n - 1$; in particular, $D_u x^n =u_n x^{n - 1} $ where $(u)$ is a given sequence of real or complex numbers subject to the restrictions $u_0 = 0$, $u_1 = 1$, $u_n \ne 0$ for $n \geqq 1$. Some of the algebraic properties of this class of polynomials are then used to study an important particular example.