In search of a higher Bochner theorem
type: Item
instance of: scholarly article
title: In search of a higher Bochner theorem
author: Emil Horozov
author: Boris Shapiro
author: Miloš Tater
published in: Journal of Approximation Theory
publication date: 2024-12-11T00:00:00Z
review text: The classical Bochner-Krall problem asks which linear ordinary differential operators possess sequences of eigenpolynomials satisfying linear recurrence relations of finite length; the classical case corresponds to 3-term recurrence relations with real coefficients subject to some extra restrictions. The authors formulate the generalized Bochner-Krall problem (GBKP).\N\NTheir main result reads: All irreducible differential operators of order 3 solving GBKP whose corresponding difference operators also have order 3 are given by the two types: \N\NType 1: \(L=a_1\partial +a_2x\partial^2+a_3x^2\partial^3+x\partial\), \(a_j\in \mathbb{C}\), \(a_3\neq 0\), \(\partial =d/dx\).\NThey generate the 2-orthogonal monic polynomials and they satisfy a 4-term recurrence relation with the standard initial conditions \(P_{-2}(x)=P_{-1}(x)=0\), \(P_0(x)=1\). \N\NType 2: \(L=a_1\partial +a_2\partial^2+a_3\partial^3+x\partial\), \(a_j\in \mathbb{C}\), \(a_3\neq 0\), generating the monic Appell polynomials. They also satisfy a 4-term recurrence relation with the standard initial condition.
reviewed by: Vladimir P. Kostov
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 33E05
zbMATH DE Number: 7957100
zbMATH Keywords: Bochner-Krall problem
zbMATH Keywords: finite recurrence relations
zbMATH Keywords: Darboux transform
zbMATH Keywords: multi-orthogonal polynomial
MaRDI profile type: Publication
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DOI: 10.1016/J.JAT.2024.106114
description: scientific article; zbMATH DE number 7957100