Recursive Procedures for Krall-Sheffer Operators

TL;DR

This paper explores the algebraic structures of Krall-Sheffer operators, providing explicit recurrence relations, raising operators, and generating functions for polynomial eigenfunctions, revealing unusual commutation relations.

Contribution

It introduces explicit recurrence relations, raising operators, and new generating functions for Krall-Sheffer operators, highlighting their algebraic properties and symmetries.

Findings

Explicit 3-level recurrence relations derived

Differential raising operators constructed

New generating functions presented

Abstract

We consider the Krall-Sheffer class of admissible, partial differential operators in the plane. We concentrate on algebraic structures, such as the role of commuting operators and symmetries. For the polynomial eigenfunctions, we give explicit forms of the the $3-$level recurrence relations and differential raising operators, which are shown to satisfy unusual commutation relations. We present new generating functions for two of the cases.