Spectral properties related to generalized complementary Romanovski-Routh polynomials

TL;DR

This paper introduces a generalized form of Complementary Romanovski-Routh polynomials with hypergeometric representation, analyzes their spectral properties, and explores their orthogonality, zero interlacing, and LU decomposition characteristics.

Contribution

It presents a new generalized class of polynomials satisfying R_{II} recurrence, analyzes their spectral and orthogonality properties, and investigates their LU decomposition and zero interlacing.

Findings

Spectral properties analyzed via tri-diagonal linear pencil.

LU decomposition reveals biorthogonality properties.

Interlacing of zeros established for the polynomials.

Abstract

Complementary Romanovski-Routh polynomials play an important role in extracting specific properties of orthogonal polynomials. In this work, a generalized form of the Complementary Romanovski-Routh polynomials (GCRR) that has the Gaussian hypergeometric representation and satisfies a particular type of recurrence called $R_{II}$ type three term recurrence relation involving two arbitrary parameters is considered. Self perturbation of GCRR polynomials leading to extracting two different types of $R_{II}$ type orthogonal polynomials are identified. Spectral properties of these resultant polynomials in terms of tri-diagonal linear pencil were analyzed. The LU decomposition of these pencil matrices provided interesting properties involving biorthogonality. Interlacing properties between the zeros of the polynomials in the discussion are established.