A quadratic functional for a third order linear difference equation

We will define a certain quadratic functional and use it to prove various results for the third order difference equation l₃y(t)=Δ³y(t-1)+p(t)Δy(t)+q(t)y(t)=0. In particular we will define kth order generlized zeroes for solutions of this equations and define (2, 1)- and (1,2)-disconjugacy of l₃y=0 on [a,b+3]. Then we will use our quadratic functional to prove sufficient conditions for (2,1)- and (1,2)-disconjugacy. We will also discuss what we call type I and II solutions of l₃y=0 and give properties of these solutions. These later results give asymptotic behavior of solutions at infinity.