Monomial and Rodrigues orthogonal polynomials on the cone

TL;DR

This paper introduces two new families of orthogonal polynomials on a cone in higher-dimensional space, providing explicit constructions and exploring their properties, including biorthogonality and generating functions.

Contribution

It presents explicit constructions of monomial and Rodrigues-type orthogonal polynomials on a cone, expanding the theory of orthogonal polynomials in higher dimensions.

Findings

Explicit formulas for monomial orthogonal polynomials on the cone.

Rodrigues-type formulas for polynomials with Laguerre or Jacobi weights.

Partial biorthogonality between the two polynomial families.

Abstract

We study two families of orthogonal polynomials with respect to the weight function $w(t)(t^2-\|x\|^2)^{\mu-\frac12}$, $\mu > -\frac 12$, on the cone $\{(x,t): \|x\| \le t, \, x \in \mathbb{R}^d, t >0\}$ in $\mathbb{R}^{d+1}$. The first family consists of monomial polynomials $\mathsf{V}_{\mathbf{k},n}(x,t) = t^{n-|\mathbf{k}|} x^\mathbf{k} + \cdots$ for $\mathbf{k} \in \mathbb{N}_0^d$ with $|\mathbf{k}| \le n$, which has the least $L^2$ norm among all polynomials of the form $t^{n-|\mathbf{k}|} x^\mathbf{k} + \mathsf{P}$ with $\deg \mathsf{P} \le n-1$, and we will provide an explicit construction for $\mathsf{V}_{\mathbf{k},n}$. The second family consists of orthogonal polynomials defined by the Rodrigues type formulas when $w$ is either the Laguerre weight or the Jacobi weight, which satisfies a generating function in both cases. The two families of polynomials are partially biorthogonal.