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Mathematics of Computation

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On Faber polynomials generated by an $ m$-star

Authors: J. Bartolomeo and Matthew He
Journal: Math. Comp. 62 (1994), 277-287
MSC: Primary 30C45; Secondary 41A58
MathSciNet review: 1203732
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Abstract: In this paper, we study the Faber polynomials $ {F_n}(z)$ generated by a regular m-star $ (m = 3,4, \ldots )$

$\displaystyle {S_m} = \{ {x{\omega ^k};0 \leq x \leq {4^{1/m}},k = 0,1, \ldots ,m - 1,{\omega ^m} = 1} \}.$

An explicit and precise expression for $ {F_n}(z)$ is obtained by computing the coefficients via a Cauchy integral formula. The location and limiting distribution of zeros of $ {F_n}(z)$ are explored. We also find a class of second-order hypergeometric differential equations satisfied by $ {F_n}(z)$. Our results extend some classical results of Chebyshev polynomials for a segment $ [ - 2,2]$ in the case when $ m = 2$.

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Keywords: Faber polynomials, m-star, zero distribution
Article copyright: © Copyright 1994 American Mathematical Society

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