On Legendre Multiplier Sequences

TL;DR

This paper characterizes Legendre multiplier sequences of various types, establishes their relationship with Hermite sequences, and explores connections to Laguerre sequences, advancing understanding of multiplier sequences in orthogonal polynomial systems.

Contribution

It provides a complete characterization of Legendre multiplier sequences and links them to Hermite and Laguerre sequences, offering new insights into their structure and relationships.

Findings

Legendre multiplier sequences are fully characterized for linear, quadratic, and geometric cases.

All Legendre multiplier sequences are also Hermite multiplier sequences.

The paper discusses the relationship between Legendre and generalized Laguerre multiplier sequences.

Abstract

In this paper we give a complete characterization of linear, quadratic, and geometric Legendre multiplier sequences. We also prove that all Legendre multiplier sequences must be Hermite multiplier sequences, and describe the relationship between the Legendre and generalized Laguerre multiplier sequences. We conclude with a list of open questions for further research.