# Polynomially Interpolated Legendre Multiplier Sequences

## TL;DR

This paper characterizes polynomially interpolated multiplier sequences for the Legendre basis, showing they have a specific quadratic form and exploring extensions and conjectures related to these sequences.

## Contribution

It provides a complete characterization of polynomially interpolated Legendre multiplier sequences and introduces new conjectures for extending these results.

## Key findings

- Multiplier sequences for Legendre basis with polynomial interpolation have the form h(k^2 + k).
- A non-trivial class of polynomial interpolations also form multiplier sequences.
- The paper states conjectures for extending these characterizations.

## Abstract

We prove that every multiplier sequence for the Legendre basis which can be interpolated by a polynomial has the form $\{h(k^2+k)\}_{k=0}^{\infty}$, where $h\in\mathbb{R}[x]$. We also prove that a non-trivial collection of polynomials of a certain form interpolate multiplier sequences for the Legendre basis, and we state conjectures on how to extend these results.