Expressing Reachability in Linear Recurrences, as Infinite Determinants and Rational Polynomial Equations

TL;DR

This paper introduces two mathematical tools involving determinants and rational polynomial sums to determine reachability of specific rational numbers in non-homogeneous linear recurrences.

Contribution

It derives a determinant-based method and an infinite rational polynomial sum to decide if a target rational can be reached in a linear recurrence.

Findings

Infinite determinant equals zero iff target rational is reachable.

Infinite sum of rational polynomials equals one iff target rational is reachable.

Provides new algebraic tools for analyzing linear recurrence reachability.

Abstract

We present two tools, which could be useful in determining whether or not a non-Homogenous Linear Recurrence can reach a desired rational. First, we derive the determinant that is equal to the ith term in a non-Homogenous Linear Recurrence. We use this to derive the infinite determinant that is zero, if and only if, the desired rational can be reached by some term in the recurrence. Second, we derive an infinite summation of rational Polynomials, such that this summation can be equal to 1, if and only if, the desired rational can be reached by some term in the recurrence.