Generalized Lucas congruences and linear $p$-schemes

TL;DR

This paper links Lucas congruences modulo p to linear p-schemes with a single state, and introduces natural generalizations, providing explicit congruences for certain integer sequences derived from Laurent polynomial constant terms.

Contribution

It establishes a connection between Lucas congruences and linear p-schemes, and generalizes Lucas congruences for sequences from Laurent polynomial constant terms.

Findings

Sequences satisfying Lucas congruences are described by linear p-schemes with one state.

Explicit generalized Lucas congruences are proved for sequences from Laurent polynomial constant terms.

The work broadens the understanding of Lucas congruences through scheme-based characterizations.

Abstract

We observe that a sequence satisfies Lucas congruences modulo $p$ if and only if its values modulo $p$ can be described by a linear $p$-scheme, as introduced by Rowland and Zeilberger, with a single state. This simple observation suggests natural generalizations of the notion of Lucas congruences. To illustrate this point, we prove explicit generalized Lucas congruences for integer sequences that can be represented as the constant terms of $P(x,y)^n Q(x,y)$ where $P$ and $Q$ are certain Laurent polynomials.