Abstract
New sets of orthogonal functions, which correspond to the first, second, third, and fourth kind Chebyshev polynomials with half-integer indexes, have been recently introduced. In this article, links of these new sets of irrational functions to the third and fourth kind Chebyshev polynomials are highlighted and their connections with the classical Chebyshev polynomials are shown.
Highlights
In the second half of the XIX Century, Pafnuty Lvovich Chebyshev introduced two sets of polynomials, presently known as the first and second kind Chebyshev polynomials, which are a polynomial version of the circular sine and cosine functions
By using the results of [16], we show the connections of the third and fourth kind pseudo-Chebyshev polynomials with the classical Chebyshev polynomials
The third and fourth kind Chebyshev polynomials have been introduced, and studied by several authors, because they can be applied in particular quadrature rules, where the singularity of the considered function appears at only one of the extrema (+1 or −1) of the integration interval
Summary
In the second half of the XIX Century, Pafnuty Lvovich Chebyshev introduced two sets of polynomials, presently known as the first and second kind Chebyshev polynomials, which are a polynomial version of the circular sine and cosine functions These polynomials have proved to be of fundamental importance in many questions of an applicative nature The real and imaginary part of the Bernoulli spirals define the Rodhonea (or Grandi) curves of fractional index, which often appear in natural shapes [15] This allows us to define two sets of functions corresponding to the first and second kind Chebyshev polynomials with fractional degree, called pseudo-Chebyshev. By using the results of [16], we show the connections of the third and fourth kind pseudo-Chebyshev polynomials with the classical Chebyshev polynomials
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