On connection coefficients, zeros and interception points of some perturbed of arbitrary order of the Chebyshev polynomials of second kind

TL;DR

This paper investigates how perturbing a coefficient in the recurrence relation of Chebyshev polynomials of the second kind affects their connection coefficients, zeros, and intersection points, for any order of perturbation.

Contribution

It provides a general method to compute connection coefficients and analyze zeros and intersection points of perturbed Chebyshev polynomials of the second kind at any perturbation order.

Findings

Derived explicit connection coefficients for perturbed polynomials.

Analyzed the impact of perturbations on zeros and intersection points.

Established relations between perturbed and original Chebyshev polynomials.

Abstract

Orthogonal polynomials satisfy a recurrence relation of order two, where appear two coefficients. If we modify one of these coefficients at a certain order, we obtain a perturbed orthogonal sequence. In this work we consider in this way some perturbed of Chebyshev polynomials of second kind and we deal with the problem of finding the connection coefficients that allow to write the perturbed sequence in terms of the original one and in terms of the canonical basis. From the connection relations obtained and from two other relations, we deduce some results about zeros and interception points of these perturbed polynomials. All the work is valid for arbitrary order of perturbation.