Reciprocal sums of differences of zeros of entire functions satisfying linear differential equations (Q1080007)

\textit{S. Ahmed} and the reviewer, SIAM J. Math. Anal. 14, 372-382 (1983; Zbl 0517.33010), gave a unifield approach to the evaluation of sums of the form \[ S_ j=\sum_{k\neq j}(z_ j-z_ k)^{-1}\quad and\quad \sum_{k}(\alpha -z_ k)^{-1}, \] where the \(z_ k\) are the (simple) zeros, satisfying \(\Sigma | z_ k| <\infty\), of a solution y of a differential equation \[ y''+P(z)y'+Q(z)y=0, \] having a singularity at \(\alpha\). The present authors extend these results to the situation where y can have multiple zeros which necessarily coincide with a singularity. Their results are applicable to certain cases of the hypergeometric equation and the modified Legendre equation.