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In search of higher Bochner theorem

Emil Horozov, Boris Shapiro, Milos Tater

Abstract

We initiate the study of a natural generalisation of the classical Bochner-Krall problem asking which linear ordinary differential operators possess sequences of eigenpolynomials satisfying linear recurrence relations of finite length; the classical case corresponds to the 3-term recurrence relations with real coefficients subject to some extra restrictions. We formulate a general conjecture and prove it in the first non-trivial case of operators of order 3.

In search of higher Bochner theorem

Abstract

We initiate the study of a natural generalisation of the classical Bochner-Krall problem asking which linear ordinary differential operators possess sequences of eigenpolynomials satisfying linear recurrence relations of finite length; the classical case corresponds to the 3-term recurrence relations with real coefficients subject to some extra restrictions. We formulate a general conjecture and prove it in the first non-trivial case of operators of order 3.

Paper Structure

This paper contains 13 sections, 14 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. On the classical problem
  3. Algebraisation and generalisation
  4. Preliminary facts
  5. The ad-condition
  6. Case when $\mathcal{L}$ is given by a linear polynomial
  7. Case $\deg a_3(x)=2 \Leftrightarrow \beta \neq 0$
  8. Case $\deg a_3(x) \leq1 \Leftrightarrow a_{32} = 0$
  9. Case when $\mathcal{L}$ is given by a quadratic polynomial
  10. Case when $\mathcal{L}$ is given by a cubic polynomial
  11. Final Remarks
  12. Appendix I. Proving Lemma \ref{['lm2.1']} by explicit computations
  13. Appendix II. Computer-aided formulas relevant for § \ref{['sec:quadr']}

Key Result

Proposition 1.1

A differential operator solves the classical Bochner-Krall problem with a positive weight function $w(x)$ if and only if it solves Problem CBK-problem and all its eigenpolynomials are real-rooted and the roots of any two consecutive eigenpolynomials are strictly interlacing.

Theorems & Definitions (31)

  • Proposition 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem A
  • Theorem B
  • Conjecture 1.7
  • Remark 1.8
  • ...and 21 more