Bivariate Generalized Shifted Gegenbauer Orthogonal System

Abstract

For $K_0,K_1\ge 0$ , $\lambda >-\left(1/2\right)$ , we examine ${\mathcal{C}}_r^{\ast \left(\lambda ,K_0,K_1\right)}\left(x\right)$ , generalized shifted Gegenbauer orthogonal polynomials, with reference to the weight $W^{\left(\lambda ,K_0,K_1\right)}\left(x\right)=\left(\left(2\lambda \mathrm{\Gamma }\left(2\lambda \right)\right)/\left(\mathrm{\Gamma }{\left(\lambda +\left(1/2\right)\right)}^2\right)\right){\left(x-x^2\right)}^{\lambda -\left(1/2\right)}\mathbb{I}\left(x\in \left(\mathrm{0,1}\right)\right)\text{d}x+K_0{\delta }_0+K_1{\delta }_1$ , where the indicator function is denoted by $\mathbb{I}\left(x\in \left(\mathrm{0,1}\right)\right)$ and ${\delta }_x$ indicates the Dirac $\delta -$ measure. Then, we construct a bivariate generalized shifted Gegenbauer orthogonal system ${\mathrm{ℭ}}_{n,r,d}^{\ast \left(\lambda ,K_0,K_1\right)}\left(u,v,w\right)$ over a triangular domain $T$ , with reference to a bivariate measure $W^{\left(\lambda ,\gamma ,K_0,K_1\right)}\left(u,v,w\right)=\left(\left(\mathrm{\Gamma }\left(2\lambda +1\right)\right)/\mathrm{\Gamma }{\left(\lambda +\left(1/2\right)\right)}^2\right)u^{\lambda -\left(1/2\right)}{\left(1-v\right)}^{\lambda -\left(1/2\right)}{\left(1-w\right)}^{\gamma -1}\mathbb{I}\left(u\in \left(\mathrm{0,1}-w\right)\right)$ $\mathbb{I}\left(w\in \left(\mathrm{0,1}\right)\right)\text{d}u\text{d}w+K_0{\delta }_0\left(u\right)+K_1{\delta }_{w-1}\left(u\right)$ , which is given explicitly in the Bézier form as ${\mathrm{ℭ}}_{n,r,d}^{\ast \left(\lambda ,K_0,K_1\right)}\left(u,v,w\right)={\displaystyle {\sum }_{i+j+k=n}{a}_{i,j,k}^{n,r,d}{\mathbb{B}}_{i,j,k}^n\left(u,v,w\right)}$ . In addition, for $d=0,\dots ,k$ , $r=\mathrm{0,1},\dots ,n$ , and $n\in \left\{0\right\}\cup \mathbb{N}$ , we write the coefficients $a_{i,j,k}^{n,r,d}$ in closed form and produce an equation that generates the coefficients recursively.