Sequences of polynomials satisfying the Pascal property

Abstract

Since one of the most important properties of binomial coefficients is the Pascal's triangle identity (referred to as the Pascal property) and since the sequence of binomial polynomials forms a regular basis for integer-valued polynomials, it is natural to ask whether the Pascal property holds in some more general setting, and what types of integer-valued polynomials possess the Pascal property. After defining the general Pascal property, a sequence of polynomials which satisfies the Pascal property is characterized with the classical case as an example. In connection with integer-valued polynomials, characterizations are derived for a sequence of polynomials which satisfies the Pascal property and also forms a regular basis of integer-valued polynomials; this is done both in a discrete valuation domain and in a Dedekind domain. Several classical cases are worked out as examples.