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For  K 0 , K 1 ≥ 0 ,  λ &gt; − 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.

Bivariate Generalized Shifted Gegenbauer Orthogonal System