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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 solutions of linear differential equations with real zeros; proof of a conjecture of Hellerstein and Rossi


Author: Franz Brüggemann
Journal: Proc. Amer. Math. Soc. 113 (1991), 371-379
MSC: Primary 34A20; Secondary 30D35, 34C10
MathSciNet review: 1057941
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Abstract: We prove the following conjecture that is due to Hellerstein and Rossi: Let $ \left\{ {{w_1}, \ldots ,{w_n}} \right\}$ be a fundamental system of

$\displaystyle Lw = {w^{(n)}} + {a_{n - 1}}(z){w^{(n - 1)}} + \cdots + {a_0}(z)w \equiv 0$

with polynomials $ {a_j}(z)(0 \leq j \leq n - 1)$. If each $ {w_k}(1 \leq k \leq n)$ has only finitely many nonreal zeros, then there exists a polynomial $ q(z)$ such that $ {u_k}: = \exp (q(z)){w_k}(1 \leq k \leq n)$ form a fundamental system of a homogeneous linear differential equation with constant coefficients.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1057941-9
PII: S 0002-9939(1991)1057941-9
Article copyright: © Copyright 1991 American Mathematical Society