Summary: In this paper we study the algebraic structure of the class of polynomials \(\{u_n \mid H_n (x)\}\) 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, \(D_u x^n =u_nx^{n-1} \) where \((u)\) is a given sequence of real or complex numbers subject to the restrictions \(u_0 = 0\), \(u_1 = 1\), \(u_n \neq 0\) for \(n \geq 1\). Some of the algebraic properties of this class of polynomials are then used to study an important particular example. For Part I, see Riv. Mat. Univ. Parma, II. Ser. 12, 47--55 (1971; Zbl 0284.33007).