Monomial and Rodrigues orthogonal polynomials on the cone

We study two families of orthogonal polynomials with respect to the weight function w(t)(t2−‖x‖2)μ−12, μ>−12, on cone {(x,t):‖x‖≤t,x∈Rd,t>0} in Rd+1. The first family consists monomial Vk,n(x,t)=tn−|k|xk+⋯ for k∈N0d |k|≤n, which has least L2 norm among all form tn−|k|xk+P deg⁡P≤n−1, and we will provide an explicit construction Vk,n. second defined by Rodrigues type formulas when w is either Laguerre or Jacobi weight, satisfies a generating both cases. are partially biorthogonal.