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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

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


Author: Li-Chien Shen
Journal: Proc. Amer. Math. Soc. 117 (1993), 1057-1061
MSC: Primary 34A20; Secondary 30D20
MathSciNet review: 1132424
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1132424-8
PII: S 0002-9939(1993)1132424-8
Article copyright: © Copyright 1993 American Mathematical Society