Sequences of polynomials satisfying the Pascal property

Authors: TUANGRAT CHAICHANA, VICHIAN LAOHAKOSOL, RATTIYA MEESA

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.

Keywords: Pascal property, integer-valued polynomial, discrete valuation domain with finite residue field, Dedekind domain

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