Fundamental Solutions for the Laplace–Beltrami Operator Defined by the Conformal Hyperbolic Metric and Jacobi Polynomials

Abstract

In this paper we study fundamental solutions for the Laplace–Beltrami operator Δαf=xnαn-2(Δf-αxn∂f∂xn), defined on smooth enough functions in R+n={(x1,…,xn)∈Rn:xn>0} . We represent explicit formulas for the fundamental solutions. Moreover, we establish fundamental solutions using Jacobi polynomials when n= 3 , 5 , 7 , ….