Donoghue $m$-functions for singular Sturm--Liouville operators

Let $\dot{A}$ be a densely defined, closed, symmetric operator in the complex, separable Hilbert space $\mathcal{H}$ with equal deficiency indices and denote by ${\mathcal{N}}_i=\ker \left(\right(\dot{A}{)}^{\ast }-i{I}_{\mathcal{H}})$, $\dim ({\mathcal{N}}_i)=k\in \mathbb{N}\cup \{\mathrm{\infty }\}$, the associated deficiency subspace of $\dot{A}$ . If $A$ denotes a self-adjoint extension of $\dot{A}$ in $\mathcal{H}$, the Donoghue $m$-operator $M_{A,$\mathcal{N}$_i}^{Do} (, \cdot ,)$in$$\mathcal{N}$_i$associatedwiththepair$(A,$\mathcal{N}$_i)$ is given by $$M_{A,{\mathcal{N}}_i}^{Do}(z)=z{I}_{{\mathcal{N}}_i}+(z^2+1)P_{{\mathcal{N}}_i}(A-z{I}_{\mathcal{H}}{)}^{-1}{P}_{{\mathcal{N}}_i}{\mid }_{{\mathcal{N}}_i},z\in \mathbb{C}\mathrm{\backslash }\mathbb{R},$$ with $I_{{\mathcal{N}}_i}$ the identity operator in ${\mathcal{N}}_i$, and $P_{{\mathcal{N}}_i}$ the orthogonal projection in $\mathcal{H}$ onto ${\mathcal{N}}_i$. Assuming the standard local integrability hypotheses on the coefficients $p, q,r$, we study all self-adjoint realizations corresponding to the differential expression $$\tau =\frac{1}{r(x)}[-\frac{d}{dx}p(x)\frac{d}{dx}+q(x)]\text{ for a.e. }x\in (a,b)\subseteq \mathbb{R}\text{,}$$ in $L^2((a,b);rdx)$, and, as the principal aim of this paper, systematically construct the associated Donoghue $m$-functions (resp., $2\times 2$ matrices) in all cases where $\tau$ is in the limit circle case at least at one interval endpoint $a$ or $b$.