$p$-Linear schemes for sequences modulo $p^r$

TL;DR

This paper introduces $p$-linear schemes to describe the behavior of certain combinatorial sequences modulo prime powers, extending the Lucas property and providing bounds on the complexity of these schemes.

Contribution

The paper constructs $p$-linear schemes for sequences modulo $p^r$ and establishes uniform upper bounds on their number of states independent of $p$.

Findings

Constructed $p$-linear schemes for sequences modulo $p^r$

Provided upper bounds on the number of states of these schemes

Extended the Lucas property to prime power moduli

Abstract

Many interesting combinatorial sequences, such as Ap\'ery numbers and Franel numbers, enjoy the so-called Lucas property modulo almost all primes $p$. Modulo prime powers $p^r$ such sequences have a more complicated behaviour which can be described by matrix versions of the Lucas property called $p$-linear schemes. They are examples of finite $p$-automata. In this paper we construct such $p$-linear schemes and give upper bounds for the number of states which, for fixed $r$, do not depend on $p$.