Recovering orthogonality from Quasi-type Kernel Polynomials using specific spectral transformations

TL;DR

This paper introduces quasi-type kernel polynomials, explores their orthogonality conditions, and presents a spectral transformation method to recover orthogonality, involving ratio expressions and continued fractions.

Contribution

It develops a new framework for quasi-type kernel polynomials and provides a method to recover orthogonality using spectral transformations and continued fractions.

Findings

Derived difference equations for quasi-type kernel polynomials

Established criteria for orthogonality conditions

Presented a spectral transformation approach involving continued fractions

Abstract

In this work, the concept of quasi-type Kernel polynomials with respect to a moment functional is introduced. Difference equation satisfied by these polynomials along with the criterion for orthogonality conditions are discussed. The process of recovering orthogonality for the linear combination of a quasi-type kernel polynomial with another orthogonal polynomial, which is identified by involving linear spectral transformation, is provided. This process involves an expression of ratio of iterated kernel polynomials. This lead to considering the limiting case of ratio of kernel polynomials involving continued fractions. Special cases of such ratios in terms of certain continued fractions are exhibited.