On a Study of a Family of Higher-Order Recurrence Relations

Abstract

In this work, group theory is applied to obtain generators of the Lie algebra for a class of rational difference equations of order $k+l$ where $k$ and $l$ are positive integers. This is followed by deriving invariants and solutions via some linear recurrences and with the introduction of some number-theoretic arithmetical functions. Furthermore, sufficient conditions for the existence of solutions for periods two and four are provided in special cases. The results in this paper generalize certain known results.