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Abstract
We consider the nth order linear difference equation$$L\sb{n}u(t) = u(t + n) + p\sb1(t)u(t + n - 1)+\cdots + p\sb{n}(t)u(t) = 0.$$Under the assumption of disconjugacy, we will prove the existence of a Polya Factorization, a Mammana Factorization, and a Trench Factorization for $L\sb{n}u(t) = 0$ and discuss some consequences of these factorizations. We will also consider the nth order linear difference equation$$l\sb{n}y(t) = L\sb{n}y(t) + q(t)y \left(t + \left\lfloor {n\over2}\right\rfloor\right) = 0.$$We will define the adjoint operator $l\sbsp{n}{*}$ of $l\sb{n}$ and give some properties relating certain solutions of $l\sb{n}y(t) = 0$ to solutions of $l\sbsp{n}{*}z(t) = 0.$ Finally, we will investigate a special case of the equation $l\sb{n}y(t) = 0.$
