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

 

A family of polynomials with concyclic zeros. II


Authors: Ronald J. Evans and Kenneth B. Stolarsky
Journal: Proc. Amer. Math. Soc. 92 (1984), 393-396
MSC: Primary 30C15; Secondary 33A10
MathSciNet review: 759660
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Abstract: Let $ {\lambda _1}, \ldots ,{\lambda _J}$ be nonzero real numbers. Expand

$\displaystyle E(z) = \prod {( - 1 + \exp {\lambda _j}z)} ,$

rewrite products of exponentials as single exponentials, and replace every $ \exp (az)$ by its approximation $ {(1 + a{n^{ - 1}}z)^n}$, where $ n \geqslant J$. The resulting polynomial has all zeros on the (possibly infinite) circle of radius $ \left\vert r \right\vert$ centered at $ - r$, where $ r = n/\sum {\lambda _j}$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0759660-9
PII: S 0002-9939(1984)0759660-9
Keywords: Concyclic zeros, exponential, exponential polynomial, linear fractional transformations, zeros of polynomials
Article copyright: © Copyright 1984 American Mathematical Society