Exceptional Laurent biorthogonal polynomials through spectral transformations of generalized eigenvalue problems

TL;DR

This paper develops a spectral transformation approach for generalized eigenvalue problems to construct exceptional Laurent biorthogonal polynomial systems, extending classical orthogonal polynomials with gaps in degree and biorthogonality.

Contribution

It introduces a novel spectral transformation framework to generate exceptional Laurent biorthogonal polynomials, including extensions of Hendriksen-van Rossum polynomials.

Findings

Constructed Laurent biorthogonal polynomial systems with degree gaps

Extended classical orthogonal polynomials to exceptional types

Demonstrated state-deletion and state-addition mechanisms

Abstract

A formulation is given for the spectral transformation of the generalized eigenvalue problem through the decomposition of the second-order differential operators. This allows us to construct some Laurent biorthogonal polynomial systems with gaps in the degree of the polynomial sequence. These correspond to an exceptional-type extension of the orthogonal polynomials, as an extension of the Laurent biorthogonal polynomials. Specifically, we construct the exceptional extension of the Hendriksen-van Rossum polynomials, which are biorthogonal analogs of the classical orthogonal polynomials. Similar to the cases of exceptional extensions of classical orthogonal polynomials, both of state-deletion and state-addition occur.