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Nonlinear recurrences related to Chebyshev polynomials

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Abstract

We use two nonlinear recurrence relations to define the same sequence of polynomials, a sequence resembling the Chebyshev polynomials of the first kind. Among other properties, we obtain results on their irreducibility and zero distribution. We then study the \(2\times 2\) Hankel determinants of these polynomials, which have interesting zero distributions. Furthermore, if these polynomials are split into two halves, then the zeros of one half lie in the interval \((-1,1)\), while those of the other half lie on the unit circle. Some further extensions and generalizations of these results are indicated.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H 4R2, Canada

    Karl Dilcher

  2. Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL, 61801, USA

    Kenneth B. Stolarsky

Authors
  1. Karl Dilcher
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  2. Kenneth B. Stolarsky
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Corresponding author

Correspondence to Karl Dilcher.

Additional information

To the memory of Marvin I. Knopp, colleague and treasured mentor.

Work supported in part by the Natural Sciences and Engineering Research Council of Canada.

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Dilcher, K., Stolarsky, K.B. Nonlinear recurrences related to Chebyshev polynomials. Ramanujan J 41, 147–169 (2016). https://doi.org/10.1007/s11139-014-9620-5

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  • DOI: https://doi.org/10.1007/s11139-014-9620-5

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