A Linear Representation for Constant Term Sequences mod $p^a$ with Applications to Uniform Recurrence
Nadav Kohen
TL;DR
The paper studies when constant-term sequences $s_n = \mathrm{ct}[P^nQ]$ modulo primes or prime powers exhibit uniform recurrence, introducing a $p$-linear representation that parallels the Rowland–Zeilberger automaton while decoupling from $Q$. It proves a sharp dichotomy: if $p$ never divides $\mathrm{ct}[P^n]$, then $s_n$ is linearly recurrent (hence uniformly recurrent); if $p$ divides some $\mathrm{ct}[P^n]$, zeros occur with density 1 and uniform recurrence fails. This dichotomy extends from primes to prime powers $p^a$ and to combinations of multiple $Q$, with a fixed-point morphism and a $p$-automatic framework underpinning the results. An application to representation theory shows that multiplicities in tensor powers of representations either all exhibit simultaneous linear recurrence modulo every $p^a$ or all have $0$-density, determined by the divisibility of a constant-term sequence. The work provides both theoretical insights and practical tools for analyzing constant-term sequences in combinatorics and representation theory.
Abstract
Many integer sequences including the Catalan numbers, Motzkin numbers, and the Apr{é}y numbers can be expressed in the form ConstantTermOf$\left[P^nQ\right]$ 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.