Effective Bounds for P-Recursive Sequences

TL;DR

This paper presents an algorithm to compute explicit upper bounds for P-recursive sequences defined by linear recurrences with polynomial coefficients, aiding in precise power series evaluation.

Contribution

It introduces a method to derive tight, explicit bounds for P-recursive sequences, enhancing the analysis of their asymptotic behavior and practical applications.

Findings

The algorithm provides bounds that match the asymptotic growth of the sequence.

Bounds are tight and applicable to power series evaluation.

The method improves understanding of P-recursive sequence behavior.

Abstract

We describe an algorithm that takes as input a complex sequence $(u_n)$ given by a linear recurrence relation with polynomial coefficients along with initial values, and outputs a simple explicit upper bound $(v_n)$ such that $|u_n| \leq v_n$ for all $n$. Generically, the bound is tight, in the sense that its asymptotic behaviour matches that of $u_n$. We discuss applications to the evaluation of power series with guaranteed precision.