Co-Related Sequences Satisfying the General Second Order Recurrence Relation

Abstract

We consider the general second order recurrence relation

$$ {A_{{n + 1}}} = a{A_n} = b{A_{{n - 1}}},\;n \in Z, $$

(1)

where a, b ∈ C are fixed, with b non-zero. For any choice of initial values A₀, A₁ ∈ C there is a unique sequence {A_n} satisfying (1). The special case of the recurrence relation (1) for which a a = - b = 1 generates the Fibonacci and Lucas numbers, where respectively A₀ = 0, A₁ = 1 and A₀ = 2, A₁ = 1. Many of the well known identities involving the Fibonacci and Lucas numbers are readily generalized to any sequence {A_n} satisfying (1). (See, for example, Vajda [2], Walton and Horadam [3].)