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.
