A series representation for the Titchmarsh-Weyl \(m\)-function (Q1206964)

The Titchmarsh-Weyl \(m\)-functions \(m_ \alpha (\lambda)\), for different values of \(\alpha\), associated with \(-y''+qy=\lambda y\) on \([0,\infty)\) and \(\cos \alpha y(0)+ \sin \alpha y'(0)=0\), are connected by the relation \(m_ \alpha (\lambda)=\{m_ \beta(\lambda) \cos(\alpha-\beta) \sin(\alpha-\beta)\}/ \{m_ \beta(\lambda) \sin(\alpha-\beta)+ \cos(\alpha-\beta)\}\), \((\lambda^{1/2}=:\alpha+i \beta)\). The author considers the particular cases \(\alpha=0\) and \(\alpha=\pi/2\).