Bivariate Generalized Shifted Gegenbauer Orthogonal System

For K0,K1≥0, λ>−1/2, we examine Cr∗λ,K0,K1x, generalized shifted Gegenbauer orthogonal polynomials, with reference to the weight Wλ,K0,K1x=2λΓ2λ/Γλ+1/22x−x2λ−1/2Ix∈0,1dx+K0δ0+K1δ1, where the indicator function is denoted by Ix∈0,1 and δx indicates the Dirac δ−measure. Then, we construct a bivariate generalized shifted Gegenbauer orthogonal system ℭn,r,d∗λ,K0,K1u,v,w over a triangular domain T, with reference to a bivariate measure Wλ,γ,K0,K1u,v,w=Γ2λ+1/Γλ+1/22uλ−1/21−vλ−1/21−wγ−1Iu∈0,1−wIw∈0,1dudw+K0δ0u+K1δw−1u, which is given explicitly in the Bézier form as ℭn,r,d∗λ,K0,K1u,v,w=∑i+j+k=nai,j,kn,r,dBi,j,knu,v,w. In addition, for d=0,…,k, r=0,1,…,n, and n∈0∪ℕ, we write the coefficients ai,j,kn,r,d in closed form and produce an equation that generates the coefficients recursively.