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Abstract
We consider the third-order linear difference equation $\ell\sb3y(t)$ = 0 on the discrete interval $\lbrack a, b + 3\rbrack $ where $\ell\sb3y(t)$ = $\Delta\sp3y(t-1) + p(t)\Delta y(t) + q(t)y(t)$ = 0. We discuss many of the interesting properties of solutions of both initial value problems and boundary value problems concerning this equation.
We discretize the ordinary differential equation $L\sb3y(x) = y\prime\prime\prime(x) + P(x)y\prime(x) + Q(x)y(x)$ = 0 in order to get our difference equation $\ell\sb3y(t)$ = O. We then show that any third order linear difference equation can be transformed into $\ell\sb3y(t)$ = 0 by an appropriate change of variables. Next, we discuss the adjoint equation $\ell\sbsp{3}{+}z(t)$ = 0, existence and uniqueness of solutions of initial value problems, the Casoratian, and the Cauchy function y(t, s).
Next we present the important concepts of generalized zeros, disconjugacy, and (1,2)- and (2,1)-pairs of zeros. We prove several technical lemmas about these concepts. We also discuss several summation formulas.
We then prove several important theorems on (1,2)- and (2,1)-disconjugacy. Then we show the equivalence of disconjugacy and Polya factorization for $\ell\sb3y(t)$ = 0. Also we prove several theorems on (1,2)- and (2,1)-disconjugacy for the adjoint equation $\ell\sbsp{3}{+}z(t)$ = 0.
Next we consider the (2,1)-boundary value problem and (2,2)-disfocality for the difference equation $\ell\sb3y(t)$ = 0. We then discuss the difference equation $\rm I\! K\sb3{\it y}(t)$ = $\Delta\lbrack r(t-1)\Delta\sp2y(t-1)\rbrack $ = 0 and find a formula for the (2,2)-Green's function of the equation with the boundary conditions y(a) = y(a $ + 1) = 0 = \Delta\sp2y(b + 1$).
Finally, we give several examples of constant coefficient equations of $\ell\sb3y(t)$ = 0. We include a discussion of the theory of roots of the third order polynomial equation $x\sp3 + ax\sp2+bx+c=0.$
