Regular polygons, Morgan-Voyce polynomials, and Chebyshev polynomials

Jorma K. Merikoski
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 2, Pages 79—87
DOI: 10.7546/nntdm.2021.27.2.79-87
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Authors and affiliations

Jorma K. Merikoski
Faculty of Information Technology and Communication Sciences, Tampere University
FI-33014 Tampere, Finland

Abstract

We say that a monic polynomial with integer coefficients is a polygomial if its each zero is obtained by squaring the edge or a diagonal of a regular n-gon with unit circumradius. We find connections of certain polygomials with Morgan-Voyce polynomials and further with Chebyshev polynomials of second kind.

Keywords

  • Regular polygons
  • Morgan-Voyce polynomials
  • Chebyshev polynomials
  • Vieta polynomials

2020 Mathematics Subject Classification

  • 11B83
  • 51M20

References

  1. André-Jeannin, R. (1994). A generalization of Morgan-Voyce polynomials. The Fibonacci Quarterly, 32, 228–231.
  2. Hoggatt, V. E., & Bicknell, M. (1974). A primer for the Fibonacci numbers: Part XIV. The Fibonacci Quarterly, 12, 147–156.
  3. Horadam, A. F. (1996). Polynomials associated with generalized Morgan-Voyce polynomials. The Fibonacci Quarterly, 34, 342–348.
  4. Horadam, A. F. (1997). A composite of Morgan-Voyce generalizations. The Fibonacci Quarterly, 35, 233–239.
  5. Horadam, A, F. (2002). Vieta polynomials. The Fibonacci Quarterly, 40, 223–232.
  6. Morgan-Voyce, A. M. (1959). Ladder networks analysis using Fibonacci numbers. IRE Transactions on Circuit Theory, 6, 321–322.
  7. Mustonen, S., Haukkanen, P., & Merikoski, J. (2014). Some polynomials associated with regular polygons. Acta Universitatis Sapientiae, Mathematica, 6, 178–193.
  8. OEIS: The On-Line Encyclopedia of Integer Sequences. Available online at http://oeis.org/.
  9. Swamy, M. N. S. (1966). Properties of the polynomials defined by Morgan-Voyce. The Fibonacci Quarterly, 4, 73–81.
  10. Swamy, M. N. S. (1968). Further properties of Morgan-Voyce polynomials. The Fibonacci Quarterly, 6, 167–175.
  11. Vaino, N. (2020). Private communication.

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Cite this paper

Merikoski, J. K. (2021). Regular polygons, Morgan-Voyce polynomials, and Chebyshev polynomials. Notes on Number Theory and Discrete Mathematics, 27(2), 79-87, doi: 10.7546/nntdm.2021.27.2.79-87.

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