Block orthogonal polynomials: II. Hermite and Laguerre standard block orthogonal polynomials

TL;DR

This paper introduces Hermite and Laguerre standard block orthogonal polynomials, extending the SBO framework to these classical polynomial families and exploring their properties for applications with linear constraints.

Contribution

It determines and investigates Hermite and Laguerre SBO polynomials, providing new bases for functional spaces with orthogonality under two scalar products.

Findings

Hermite SBO polynomials are orthogonal with respect to Hermite-based scalar products.

Laguerre SBO polynomials are orthogonal with respect to Laguerre-based scalar products.

These polynomials are useful for problems with linear constraints in functional spaces.

Abstract

The standard block orthogonal (SBO) polynomials $P_{i;n}(x), 0\le i\le n$ are real polynomials of degree $n$ which are orthogonal with respect to a first Euclidean scalar product to polynomials of degree less than $i$. In addition, they are mutually orthogonal with respect to a second Euclidean scalar product. Applying the general results obtained in a previous paper, we determine and investigate these polynomials when the first scalar product corresponds to Hermite (resp. Laguerre) polynomials. These new sets of polynomials, we call Hermite (resp. Laguerre) SBO polynomials, provide a basis of functional spaces well-suited for some applications requiring to take into account special linear constraints which can be recast into an Euclidean orthogonality relation.