On connection coefficients of some perturbed of arbitrary order of the Chebyshev polynomials of second kind

ABSTRACT

Orthogonal polynomials satisfy a recurrence relation of order two defined by two sequences of coefficients. If we modify one of these recurrence coefficients at a certain order, we obtain the so-called perturbed orthogonal sequence. In this work, we analyse perturbed Chebyshev polynomials of second kind and we deal with the problem of finding the connection coefficients that allow us to write the perturbed sequence in terms of the original one and in terms of the canonical basis. From the connection coefficients obtained, we derive some results about zeros at the origin. The analysis is valid for arbitrary order of perturbation.

KEYWORDS:

• Chebyshev polynomials
• perturbed orthogonal polynomials
• connection coefficients
• zeros

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 33C45
• 33D45
• 42C05

Acknowledgments

I am very grateful to Pascal Maroni for several discussions during the development of this work. The author would like to thank the referee for his comments in order to improve the presentation of this article.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 CCOP is written in the Mathematica${}^{\circledR }$ language and is available in the library Numeralgo of Netlib (http://www.netlib.org/numeralgo/) as na34 package.

Additional information

Funding

The author was partially supported by CMUP (UID/MAT/00144/2019), which is funded by Fundação para a Ciência e a Tecnologia (FCT) (Portugal) with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020.