A Congruence Relation for a Linear Recursive Sequence of Arbitrary Order

Abstract

We consider the sequence {T n } 0 ∞ defined by $$ {T_{n + m + 1}} = \sum\limits_{r = 0}^m {{a_r}\quad {T_{n + r}},\;n \ge 0} $$ (1) , with initial conditions $$ {T_r} = {c_r},\quad 0 \le r \le m $$ , where the a r and c r are integers and a0 ≠ 0. (If the c r are all zero, {T n }∞ 0 becomes the null sequence. In this case Theorems 1 and 2 below are trivial.) In (1) m ≥ 0 is a fixed integer. We referee to (1) as an (m+1)th order recurrence relation or an (m+1)th order difference equation. Thus {T n } is an integer sequence. The purpose of our present paper is to generalize results which we obtained [2] for a sequence {T n } defined by a second order recurrence relation (m = 1 in (1)), the Fibonacci and Lucas sequences being important special cases. (The case m = 0 is trivial.)