An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences

Abstract

In 1939, Sheffer published “Some properties of polynomial sets of type zero”, which has been regarded as an indispensable paper in the theory of orthogonal polynomials. Therein, Sheffer basically proved that every polynomial sequence can be classified as belonging to exactly one type. In addition to various interesting and important relations, Sheffer’s most influential results pertained to completely characterizing all of the polynomial sequences of the most basic type, called A-type 0, and subsequently establishing which of these sets were also orthogonal. However, Sheffer’s elegant analysis relied heavily on several characterization theorems. In this work, we show all of the Sheffer A-type 0 orthogonal polynomial sequences can be characterized by using only the generating function that defines this class and a monic three-term recurrence relation.