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

Authors : Julius Fergy Tiongson RABAGO

Pages : 923-933

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Publication Date : 2014-12-01

Article Type : Research Paper

Abstract :We say that winsert ignore into journalissuearticles values(x);: R → C is a solution to a second-order linear recurrent homogeneous differential equation with period k insert ignore into journalissuearticles values(k ∈ N);, if it satisfies a homogeneous differential equation of the form w insert ignore into journalissuearticles values(2k); insert ignore into journalissuearticles values(x); = pw insert ignore into journalissuearticles values(k); insert ignore into journalissuearticles values(x); + qwinsert ignore into journalissuearticles values(x);, ∀x ∈ R, where p, q ∈ R + and w insert ignore into journalissuearticles values(k); insert ignore into journalissuearticles values(x); is the k th derivative of winsert ignore into journalissuearticles values(x); with respect to x. On the other hand, winsert ignore into journalissuearticles values(x); is a solution to an odd second-order linear recurrent homogeneous differential equation with period k if it satisfies w insert ignore into journalissuearticles values(2k); insert ignore into journalissuearticles values(x); = −pw insert ignore into journalissuearticles values(k); insert ignore into journalissuearticles values(x); + qwinsert ignore into journalissuearticles values(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 winsert ignore into journalissuearticles values(x); is the general solution to a second-order linear recurrent homogeneous differential equation with period k insert ignore into journalissuearticles values(resp. odd second-order linear recurrent homogeneous differential equation with period k);, then the limit of the quotient w insert ignore into journalissuearticles values(insert ignore into journalissuearticles values(n+1);k); insert ignore into journalissuearticles values(x);/winsert ignore into journalissuearticles values(n); insert ignore into journalissuearticles values(x); as n tends to infinity exists and is equal to the positive insert ignore into journalissuearticles values(resp. negative); dominant root of the quadratic equation x 2 − px − q = 0 as x increases insert ignore into journalissuearticles values(resp. decreases); without bound.
Keywords : Homogenous differential equations, second order linear recurrence sequences, solutions