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) $\ge 2$. We characterise the dimension of the quotient space $D(T_{\max })\mathrm{/}D(T_{\min })$ 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}] {equation*} where $1\le n\le$ (order of the expression + 1); here $[fg](\mathrm{\infty })=\underset{x\to \mathrm{\infty }}{\lim }[fg](x)$, 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 $\mathrm{\infty }$ if and only if $[fg](\mathrm{\infty })=0$ for every $f,g\in$ the maximal domain $\mathrm{\Delta }$ associated with $M$.