Solving Third-Order Linear Recurrence Relations with Applications to Number Theory and Combinatorics

Abstract

In this paper we develop a novel matrix method for solving linear recurrence relations and present explicit formulae for the general solution of the third-order linear homogeneous recurrence relations with variable coefficients. We obtain a summatory formula for the general solution of the recurrence relation in the special case. We review some known results and then consider some particular cases of the recurrence and examples with applications to combinatorics, especially to number sequences and polynomials. Finally, we briefly discuss further generalization of the method for higher order linear recurrence relations.