On the zeros of the solutions of $y”+P(z)y=0$ where $P(z)$ is a polynomial

Abstract:

Let $\{ {z_n}\}$ be the nonzero zeros of the differential equation $y'' + P(z)y = 0$, where $P(z) = {a_0} + {a_1}z + {a_2}{z^2} + \cdots + {a_N}{z^N}$, and let ${c_k} = \sum \nolimits _{n = 1}^\infty {1/z_n^k\;{\text {for}}\;k \geqslant [N/2] + 2}$. We show that ${c_k}$ is a rational function of ${a_n},\;n = 0,1,2, \ldots ,N$; futhermore, the successive ${c_k}$ can be computed from previous ${c_k}$’s by a simple recurrence relation.