Monomial and Rodrigues orthogonal polynomials on the cone

Abstract

We study two families of orthogonal polynomials with respect to the weight function $w(t)(t^2-\Vert x{\Vert }^2{)}^{\mu -\frac{1}{2}}$, $\mu >-\frac{1}{2}$, on the cone $\{(x,t):\Vert x\Vert \le t,x\in {\mathbb{R}}^d,t>0\}$ 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 $°\mathsf{P}\le n-1$, 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.

24 pp