P-adic Valuations and K-regular Sequences

Abstract A sequence is said to be k-automatic if the nth term of this sequence is generated by a finite state machine with n in base k as input. Regular sequences were first defined by Allouche and Shallit as a generalization of automatic sequences. Given a prime p and a polynomial f(x) ∈ Qp[x], we consider the sequence {vp(f(n))}n=0, where vp is the p-adic valuation. We show that this sequence is p-regular if and only if f(x) factors into a product of polynomials, one of which has no roots in Zp, the other which factors into linear polynomials over Q. This answers a question of Allouche and Shallit.