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Recurrence relations of Exceptional Laurent biorthogonal polynomials

Yu Luo, Satoshi Tsujimoto, Hao Yang

Abstract

Exceptional extensions of a class of Laurent biorthogonal polynomials (the so-called Hendriksen-van Rossum polynomials) have been presented by the authors recently. This is achieved through Darboux transformations of generalized eigenvalue problems. In this paper, we discuss the recurrence relations satisfied by these exceptional Laurent biorthogonal polynomials and provide a type of recurrence relations with $3l_0+4$ terms explicitly, where the parameter $l_0$ corresponds to the degree of the polynomial part in the seed function used in the Darboux transformation. In the proof of these recurrence relations, the backward operator which maps an exceptional polynomial into a classical one plays a significant role.

Recurrence relations of Exceptional Laurent biorthogonal polynomials

Abstract

Exceptional extensions of a class of Laurent biorthogonal polynomials (the so-called Hendriksen-van Rossum polynomials) have been presented by the authors recently. This is achieved through Darboux transformations of generalized eigenvalue problems. In this paper, we discuss the recurrence relations satisfied by these exceptional Laurent biorthogonal polynomials and provide a type of recurrence relations with terms explicitly, where the parameter corresponds to the degree of the polynomial part in the seed function used in the Darboux transformation. In the proof of these recurrence relations, the backward operator which maps an exceptional polynomial into a classical one plays a significant role.
Paper Structure (9 sections, 22 theorems, 193 equations)

This paper contains 9 sections, 22 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Darboux transformation of a Generalized eigenvalue problem
  3. quasi-Laurent-polynomial eigenfunctions
  4. Hendriksen-van Rossum polynomials
  5. Expansion formulas related to HR polynomials
  6. Exceptional Hendriksen-van Rossum polynomials
  7. Proofs of Theorem \ref{['thm_main_1']} and Theorem \ref{['thm_main_2']}
  8. Recurrence relations of X-HR polynomials: concrete examples
  9. Concluding remarks

Key Result

Theorem 1.1

If $l_0\geq 1$ and $n\geq k$, then the exceptional HR polynomials $\{P^{(j_0,l_0,n)}(z)\}_n$, $j_0\in\{1,2,3,4\}$, satisfy the following recurrence relations: where $q^{(j_0)}_{l_0}(z)$ is a polynomial of degree $l_0+1$ defined by (qpi), $a_l^{(j_0,l_0,n)}$, $l=0,\ldots,k$, and $b_j^{(j_0,l_0,n)}$, $j=0,\ldots,n+l_0+1$, are constants.

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: XHR
  • Lemma 2.2: XHR
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • proof
  • ...and 26 more