Bivariate Generalized Shifted Gegenbauer Orthogonal System

For K 0 , K 1 ≥ 0 , λ > − 1 / 2 , we examine C r ∗ λ , K 0 , K 1 x , generalized shifted Gegenbauer orthogonal polynomials, with reference to the weight W λ , K 0 , K 1 x = 2 λ Γ 2 λ / Γ λ + 1 / 2 2 x − x 2 λ − 1 / 2 I x ∈ 0,1 d x + K 0 δ 0 + K 1 δ 1 , where the indicator function is denoted by I x ∈ 0,1 and δ x indicates the Dirac δ − measure. Then, we construct a bivariate generalized shifted Gegenbauer orthogonal system ℭ n , r , d ∗ λ , K 0 , K 1 u , v , w over a triangular domain T , with reference to a bivariate measure W λ , γ , K 0 , K 1 u , v , w = Γ 2 λ + 1 / Γ λ + 1 / 2 2 u λ − 1 / 2 1 − v λ − 1 / 2 1 − w γ − 1 I u ∈ 0,1 − w I w ∈ 0,1 d u d w + K 0 δ 0 u + K 1 δ w − 1 u , which is given explicitly in the Bézier form as ℭ n , r , d ∗ λ , K 0 , K 1 u , v , w = ∑ i + j + k = n a i , j , k n , r , d B i , j , k n u , v , w . In addition, for d = 0 , … , k , r = 0,1 , … , n , and n ∈ 0 ∪ ℕ , we write the coefficients a i , j , k n , r , d in closed form and produce an equation that generates the coefficients recursively.