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

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.