On connection coefficients, zeros and interception points of some perturbed of arbitrary order of the Chebyshev polynomials of second kind
This work provides theoretical results for a specific class of orthogonal polynomials, which is incremental for the field of special functions.
The authors derive connection coefficients for perturbed Chebyshev polynomials of the second kind, enabling expression of the perturbed sequence in terms of the original polynomials and the canonical basis. They then use these relations to deduce properties about zeros and interception points, with results valid for arbitrary perturbation order.
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.