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Abstract
The nth order linear difference equation Pu(m) = 0 on an integer interval I = (a,b) = $\{a,a$ + 1,...,b$\}$ is considered, where $Pu(m)$ = $\sum\sbsp{i=0}{n}\ \alpha\sb{i}$(m)u(m + $i$) with the assumptions $\alpha\sb{n}$(m) = 1 and ($-$1)$\sp{n}\alpha\sb0$(m) $>$ 0 on I. Solutions for $Pu(m)$ = 0 are defined on I$\sp{n}$ = (a, b + n).
Necessary and sufficient conditions for $Pu(m)$ = 0 to be disconjugate on I$\sp{n}$ in terms of determinants involving the coefficients $\alpha\sb{i}$(m) were given by Hartman. Here, necessary and sufficient conditions for $Pu(m)$ = 0 to be right (l,n $-$ l)-disconjugate and left (l,n $-$ l)-disconjugate on I$\sp{n}$ are given in terms of determinants involving the coefficients. These results lead to an improvement of Hartman's with regard to sufficient conditions for $Pu(m)$ = 0 to be disconjugate. These are improved further by showing the conditions need only be checked at the initial point of I. Further, the conditions for right (l, n $-$ l)-disconjugacy and left (l, n $-$ l)-disconjugacy show these two concepts are equivalent.
Next, the concept of disfocality of $Pu(m)$ = 0 on I$\sp{n}$ is considered. Necessary and sufficient conditions for the second order linear difference equation to be right disfocal, similar to those obtained for disconjugacy, are given. A definition for $Pu(m)$ = 0 to be right $\rho\sb{l}$-disfocal on I$\sp{n}$ is given, where l $\in\ \{$1,...,n $-$ 1$\}$. Necessary conditions for $Pu(m)$ = 0 to be right $\rho\sb{l}$-disfocal are given in terms of the coefficients. In the special case when l = n $-$ 1, necessary and sufficient conditions for $Pu(m)$ = 0 to be right $\rho\sb{n-1}$-disfocal are given.
With the assumption that $Pu(m)$ = 0 is right $\rho\sb{l}$-disfocal on I$\sp{n}$, consequences of the discrete analogue of Rolle's Theorem are given for a solution of a boundary value problem for $Pu(m)$ = 0.
Finally, for k $\in\ \{$0,1,...,n $-$ 1$\}$, an adjoint is defined for the linear difference equation L$\sb{k}y(m)$ $\equiv\ \Delta\sp{n}y(m - k) + q(m)y(m) = 0$ on ($a + k, b + k$). If k = 0, 1 + ($-$1)$\sp{n}q(m)$ is assumed to be positive. A Lagrange bracket function is defined, and a Lagrange identity for L$\sb{k}$ and its adjoint is given. Relationships are given for principal solutions of $L\sb{k}y(m) = 0$ and its adjoint equation.
