A Congruence Relation for a Linear Recursive Sequence of Arbitrary Order

H. T. Freitag and G. M. Phillips

A chapter in Applications of Fibonacci Numbers, 1988, pp 39-44 from Springer

Abstract: Abstract We consider the sequence {T n } 0 ∞ defined by (1) $$ {T_{n + m + 1}} = \sum\limits_{r = 0}^m {{a_r}\quad {T_{n + r}},\;n \ge 0} $$ , 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.)

Keywords: Characteristic Polynomial; Congruence Relation; Multinomial Coefficient; Important Special Case; Integer Sequence (search for similar items in EconPapers)
Date: 1988
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DOI: 10.1007/978-94-015-7801-1_5

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