Jacobi-weighted orthogonal polynomials on triangular domains

We construct Jacobi-weighted orthogonal polynomials 𝒫 n , r ( α , β , γ ) ( u , v , w ) , α , β , γ > − 1 , α + β + γ = 0 , on the triangular domain T . We show that these polynomials 𝒫 n , r ( α , β , γ ) ( u , v , w ) over the triangular domain T satisfy the following properties: 𝒫 n , r ( α , β , γ ) ( u , v , w ) ∈ ℒ n , n ≥ 1 , r = 0 , 1 , … , n , and 𝒫 n , r ( α , β , γ ) ( u , v , w ) ⊥ 𝒫 n , s ( α , β , γ ) ( u , v , w ) for r ≠ s . And hence, 𝒫 n , r ( α , β , γ ) ( u , v , w ) , n = 0 , 1 , 2 , … , r = 0 , 1 , … , n form an orthogonal system over the triangular domain T with respect to the Jacobi weight function. These Jacobi-weighted orthogonal polynomials on triangular domains are given in Bernstein basis form and thus preserve many properties of the Bernstein polynomial basis.