A closed formula for linear recurrences with constant coefficients

TL;DR

This paper derives a closed-form formula for the polynomial coefficients in the general solution of linear recurrences with constant coefficients, enhancing understanding of their structure.

Contribution

It provides a new explicit formula for the polynomial coefficients in the solutions of linear recurrences, extending previous known results.

Findings

Closed-form expressions for polynomial coefficients are derived.

The formula applies to any linear recurrence with constant coefficients.

This advances the theoretical understanding of recurrence solutions.

Abstract

Given a linear recurrence of the form $c_n=a_1c_{n-1}+\cdots+a_j c_{n-j}$, it is well-known that $c_n=\sum_{r}p_r(n)r^n$, where the sum is taken over the set of characteristic roots and each $p_r(n)$ is some polynomial. We give a closed formula for the coefficients of each polynomial $p_r(n)$ for any linear recurrence of this form.