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 {An} 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 {An} satisfying (1). (See, for example, Vajda [2], Walton and Horadam [3].)