Monomial and Rodrigues orthogonal polynomials on the cone

We study two families of orthogonal polynomials with respect to the weight function w(t)(t(2) - llxll(2))mu-1/2, mu > -1/2, on the cone {(x, t) : llxll <= t, x E R-d, t > 0} in Rd+1. The first family consists of monomial polynomials V-k,V-n(x, t) = t(n-|k|)x(k) + center dot center dot center dot for k is an element of N-0(d) with |k| <= n, which has the least L-2 norm among all polynomials of the form t(n-|k|)x(k)+P with deg P <= n -1, and we will provide an explicit construction for V-k,V-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. (c) 2022 Elsevier Inc. All rights reserved.