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New tools for the study of Bochner differential operators

L. M. Anguas, D. Barrios Rolanía

Abstract

A sequence $\{δ_n^{(k)}\}$ associated to a Bochner differential operator is introduced as an effective tool to study this kind of operators. Some properties of this sequence are proven and used to deduce that a particular operator leads to solutions of a bispectral problem. In addition, the inverse problem is studied; that is, given a sequence $\{λ_n\}$ of complex numbers and a sequence $\{P_n\}$ of polynomials with complex coefficients, $°{P_n}=n$, we find a necessary and sufficient condition for the existence of a Bochner differential operator that has those sequences as eigenvalues and eigenpolynomials, respectively. The mentioned condition also depends on $\{δ_n^{(k)}\}$.

New tools for the study of Bochner differential operators

Abstract

A sequence associated to a Bochner differential operator is introduced as an effective tool to study this kind of operators. Some properties of this sequence are proven and used to deduce that a particular operator leads to solutions of a bispectral problem. In addition, the inverse problem is studied; that is, given a sequence of complex numbers and a sequence of polynomials with complex coefficients, , we find a necessary and sufficient condition for the existence of a Bochner differential operator that has those sequences as eigenvalues and eigenpolynomials, respectively. The mentioned condition also depends on .

Paper Structure

This paper contains 6 sections, 11 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Direct problem
  4. Example of use of $\{\delta_n^{(k)}\}$ for a particular Bochner differential operator
  5. Inverse problem
  6. Conclusions and future work

Key Result

Theorem 1

Given two sequences $\{a_{n,i}\}, i=0,1,\dots,n,\, n\in \mathbb{N}$, and $\{\delta_n^{(k)}\},\ n=0,1,\dots,\ k=0,1,\dots,n,$ the following connections are equivalent

Theorems & Definitions (19)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 2
  • Theorem 3
  • proof
  • ...and 9 more