Reciprocal Power Sums of Differences of Zeros of Special Functions

We derive formulas for sums of the form \[ \sum_{\begin{subarray}{l} k - 1 \\ k \ne j \end{subarray}} ^\infty {\left( {z_j - z_k } \right)^{ - m} } \quad (m,j = 1,2,3, \cdots ),\] where $\{ z_k \} $ is the (finite or infinite) sequence of (complex) zeros of an appropriate solution of a second order linear differential equation. In special cases our results reduce to some of those obtained by T. J. Stieltjes, F. Calogero, S. Ahmed, M. L. Mehta, K. M. Case and others.