Growth and Form

Volume 1, Issue 1, 2020, Pages 20 - 32

Chebyshev Polynomials, Rhodonea Curves and Pseudo-Chebyshev Functions. A Survey

Authors
Paolo Emilio Ricci*
Department of Mathematics, International Telematic University UniNettuno, Corso Vittorio Emanuele II, 39, Roma 00186, Italy
Corresponding Author
Paolo Emilio Ricci
Received 17 December 2019, Accepted 20 December 2019, Available Online 23 February 2020.
DOI
https://doi.org/10.2991/gaf.k.200124.005How to use a DOI?
Keywords
Spirals; Grandi curves; pseudo-Chebyshev functions; recurrence relations; differential equations; orthogonality properties
Abstract

In recent works, starting from the complex Bernoulli spiral and the Grandi roses, sets of irrational functions have been introduced and studied that extend to the fractional degree the polynomials of Chebyshev of the first, second, third and fourth kind. The functions thus obtained are therefore called pseudo-Chebyshev. This article presents a review of the elementary properties of these functions, with the aim of making the topic accessible to a wider audience of readers. The subject is presented as follows. In Section 2 a review of spiral curves is given. In Section 3 the main properties of the classical Chebyshev polynomials are recalled. The Grandi (Rhodonea) curves and possible extensions are introduced in Section 4, and a method for deriving new curves, changing cartesian into polar coordinates, is touched on. The possibility to consider the Grandi curves even for rational indexes allows to introduce in Section 5 the pseudo-Chebyshev functions, which are derived from the Chebyshev polynomials assuming rational values for their degree. The main properties of these functions are shown, including recursions and differential equations. In particular, the case of half-integer degree is examined in Section 6 since, in this case, the pseudo-Chebyshev functions verify even the orthogonality property. As a consequence, new system of irrational orthogonal functions are introduced.

Copyright
© 2020 The Authors. Published by Atlantis Press SARL
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
Growth and Form
Volume-Issue
1 - 1
Pages
20 - 32
Publication Date
2020/02/23
ISSN (Online)
2589-8426
DOI
https://doi.org/10.2991/gaf.k.200124.005How to use a DOI?
Copyright
© 2020 The Authors. Published by Atlantis Press SARL
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Paolo Emilio Ricci
PY  - 2020
DA  - 2020/02/23
TI  - Chebyshev Polynomials, Rhodonea Curves and Pseudo-Chebyshev Functions. A Survey
JO  - Growth and Form
SP  - 20
EP  - 32
VL  - 1
IS  - 1
SN  - 2589-8426
UR  - https://doi.org/10.2991/gaf.k.200124.005
DO  - https://doi.org/10.2991/gaf.k.200124.005
ID  - Ricci2020
ER  -