On the Convergence of Quotients of Some Recursive Sequences

Abstract

The aim of this paper is to investigate the behavior of quotients of sequences w _n defined by the linear homogeneous recurrence relation

$$w_{n + 2} = pw_{n + 1} - qw_n$$

(1)

with arbitrary initial values w ₀ = a, w ₁ = b belonging to a normed field K, and p, q ∈ K. We shall write w _n (a,b; p,q) when it swill be necessary to specify the initial data and parameters. In [3], Horadam has studied the properties of these sequences. If w _n ≠ 0, ∀n ∈ N, then the sequence of quotien1ts

$$Q_n (a,b;p,q) = \frac{{w_{n + 1} }} {{w_n }}$$

(2.1)

is defined in K. If the field K is normed and α, β are the roots of the polynomial x ² – px + q, with ∥ α ∥ > ∥ β ∥ then the sequences of Q _n ’s, where it is defined, converges, and

$$ \mathop{{\lim }}\limits_{{n \to \infty }} {Q_n} = \alpha. $$

(3)