On second-order linear recurrent homogeneous differential equations with period k

Abstract

We say that w(x): R → C is a solution to a second-order linear recurrent homogeneous differential equation with period k (k ∈ N), if it satisfies a homogeneous differential equation of the form w (2k) (x) = pw (k) (x) + qw(x), ∀x ∈ R, where p, q ∈ R + and w (k) (x) is the k th derivative of w(x) with respect to x. On the other hand, w(x) is a solution to an odd second-order linear recurrent homogeneous differential equation with period k if it satisfies w (2k) (x) = −pw (k) (x) + qw(x), ∀x ∈ R. In the present paper, we give some properties of the solutions of differential equations of these types. We also show that if w(x) is the general solution to a second-order linear recurrent homogeneous differential equation with period k (resp. odd second-order linear recurrent homogeneous differential equation with period k), then the limit of the quotient w ((n+1)k) (x)/w(n) (x) as n tends to infinity exists and is equal to the positive (resp. negative) dominant root of the quadratic equation x 2 − px − q = 0 as x increases (resp. decreases) without bound.