On the limit-classifications of even and odd-order formally symmetric differential expressions

In this paper we consider the formally symmetric differential expression $M[\cdot]$ of any order (odd or even) $\geq 2$. We characterise the dimension of the quotient space $D(T_{\max})/D(T_{\min})$ associated with $M[\cdot]$ in terms of the behaviour of the determinants {equation*} \det\limits_{r,s\in {\bf N}_{n}} [[f_{r}g_{s}](\infty)] {equation*} where $1\leq n\leq$ (order of the expression + 1); here $[fg](\infty) = \lim\limits_{x\to\infty}[fg](x)$, where $[fg](x)$ is the sesquilinear form in $f$ and $g$ associated with $M$. These results generalise the well-known theorem that $M$ is in the limit-point case at $\infty$ if and only if $[fg](\infty) = 0$ for every $f,g\in$ the maximal domain $\Delta$ associated with $M$.