Asymptotic valuations of sequences satisfying first order recurrences

TL;DR

This paper investigates the long-term behavior of the p-adic valuation of sequences defined by first order recurrence relations involving polynomials, under specific root derivative conditions.

Contribution

It provides a detailed analysis of the asymptotic p-adic valuation of sequences satisfying first order recurrences with polynomial coefficients, extending understanding of their valuation growth.

Findings

Asymptotic formulas for p-adic valuations are derived.

Conditions on polynomial roots influence valuation behavior.

Results apply to sequences with roots having nonvanishing derivatives.

Abstract

Let t[n] be a sequence that satisfies a first order homogeneous recurrence t[n] = Q[n]*t[n-1], where Q is a polynomial with integer coefficients. The asymptotic behavior of the p-adic valuation of t[n] is described under the assumption that all the roots of Q in Z/pZ have nonvanishing derivative.