Bivariate Generalized Shifted Gegenbauer Orthogonal System

For K-0, K-1 >= 0,. > - (1/2), we examine C*(r) ((lambda,K0,K1)) (x), generalized shifted Gegenbauer orthogonal polynomials, with reference to the weight W-(lambda,W-K0,W-K1)(x). ((2 lambda Gamma(2 lambda))/(Gamma(lambda + (1/2))(2)))( x - x(2))(lambda-(1/2)) I(x epsilon (0, 1))dx + K-0 delta(0) + K-1 delta(1), where the indicator function is denoted by I(x epsilon (0, 1)) and delta x indicates the Dirac delta-measure. Then, we construct a bivariate generalized shifted Gegenbauer orthogonal system C*((lambda, K0x,K1))(n,r,d) (u, v, w) over a triangular domain T, with reference to a bivariate measure W-(lambda,W-K0,W-K1)(u, v, w). ((Gamma(2 lambda + 1))/Gamma(lambda + (1/2))(2))u(lambda-(1/2))(1 - v)lambda(-(1/2)) (1 - w)Ic-1(u subset of (0, 1 - w)) I(w subset of (0, 1))dudw + K-0 delta(0)(u) +K-1 delta(w-1)(u), which is given explicitly in the B ' ezier form as C*((lambda, K0x,K1))(n,r,d) (u, v, w) Sigma(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 epsilon {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.