A Linear Representation for Constant Term Sequences mod \(p^a\) with Applications to Uniform Recurrence

Many integer sequences including the Catalan numbers, Motzkin numbers, and the Apéry numbers can be expressed in the form ConstantTermOf[\(P^nQ\)] for Laurent polynomials \(P\) and \(Q\). These are often called “constant term sequences”. In this paper, we characterize the prime powers, \(p^a\), for which sequences of this form modulo \(p^a\), and others built out of these sequences, are uniformly recurrent. For all other prime powers, we show that the frequency of 0 is 1. This is accomplished by introducing a novel linear representation of constant term sequences modulo \(p^a\), which is of independent interest.

Published here.