Groups, conics and recurrence relations

Abstract

In this paper we explore some of the geometry that lies behind the real linear, second order, constant coefficient, recurrence relation (1) where a and b are real numbers. Readers will be familiar with the standard method of solving this relation, and, to avoid trivial cases, we shall assume that ab ≠ 0. The auxiliary equation of t2 = at + b of (1) has two (possibly complex) solutions and the most general solution of (1) is given by (i) when are real and distinct;(ii) when (iii).