Orthogonal Polynomials Associated with Some Jacobi-Type Pencils

Zagorodnyuk, Sergey M.

doi:10.1007/s11253-017-1300-3

Cited by 7 publications

(15 citation statements)

References 8 publications

(9 reference statements)

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“…Let J 3 be a Jacobi matrix and J 5 be a semi-infinite real symmetric five-diagonal matrix with positive numbers on the second subdiagonal. A set Θ ¼ J 3 , J 5 , α, β ðÞ , where α > 0, β ∈ , is said to be a Jacobi-type pencil (of matrices) [14]. With a Jacobi-type pencil of matrices Θ one associates a system of polynomials p n λ ðÞ ÈÉ ∞ n¼0 , which satisfies the following relations:…”

Section: Pencils J 5 à Xj 3 and Orthogonal Polynomialsmentioning

confidence: 99%

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Pencils of Semi-Infinite Matrices and Orthogonal Polynomials

Zagorodnyuk¹

2023

Matrix Theory - Classics and Advances

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Semi-infinite matrices, generalized eigenvalue problems, and orthogonal polynomials are closely related subjects. They connect different domains in mathematics—matrix theory, operator theory, analysis, differential equations, etc. The classical examples are Jacobi and Hessenberg matrices, which lead to orthogonal polynomials on the real line (OPRL) and orthogonal polynomials on the unit circle (OPUC). Recently there turned out that pencils (i.e., operator polynomials) of semi-infinite matrices are related to various orthogonal systems of functions. Our aim here is to survey this increasing subject. We are mostly interested in pencils of symmetric semi-infinite matrices. The corresponding polynomials are defined as generalized eigenvectors of the pencil. These polynomials possess special orthogonality relations. They have physical and mathematical applications that will be discussed. Examples show that there is an unclarified relation to Sobolev orthogonal polynomials. This intriguing connection is a challenge for further investigations.

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Section: Pencils J 5 à Xj 3 and Orthogonal Polynomialsmentioning

confidence: 99%

“…where J 3 is a Jacobi matrix, and J 5 is a real symmetric semi-infinite five-diagonal matrix with positive numbers on the second subdiagonal, see ref. [14]. These polynomials contain OPRL as a proper subclass.…”

Section: Introductionmentioning

confidence: 99%

Pencils of Semi-Infinite Matrices and Orthogonal Polynomials

Zagorodnyuk¹

2023

Matrix Theory - Classics and Advances

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“…In this section we recall basic definitions and results from [15], [16] which will be used later. Set u n := J 3 e n = a n−1 e n−1 + b n e n + a n e n+1 ,…”

Section: Preliminariesmentioning

confidence: 99%

Difference equations related to Jacobi-type pencils

Zagorodnyuk

2018

Journal of Difference Equations and Applications

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“…In this section, for the convenience of the reader, we recall basic definitions and results from [16]. Then we state the direct and inverse spectral problems for the Jacobi-type pencils.…”

Section: Preliminariesmentioning

confidence: 99%

“…(1.3). This Jacobi type pencil was considered in [16] and explicit formulas for the associated polynomials p n (λ) were obtained.…”

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confidence: 99%

The Inverse Spectral Problem for Jacobi-Type Pencils

Zagorodnyuk¹

2017

SIGMA

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Abstract. In this paper we study the inverse spectral problem for Jacobi-type pencils. By a Jacobi-type pencil we mean the following pencil J 5 − λJ 3 , where J 3 is a Jacobi matrix and J 5 is a semi-infinite real symmetric five-diagonal matrix with positive numbers on the second subdiagonal. In the case of a special perturbation of orthogonal polynomials on a finite interval the corresponding spectral function takes an explicit form.

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Orthogonal Polynomials Associated with Some Jacobi-Type Pencils

Cited by 7 publications

References 8 publications

Pencils of Semi-Infinite Matrices and Orthogonal Polynomials

Pencils of Semi-Infinite Matrices and Orthogonal Polynomials

Difference equations related to Jacobi-type pencils

The Inverse Spectral Problem for Jacobi-Type Pencils

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