We did define Jacobi-weighted orthogonal polynomials on triangular domains;

ABSTRACT We construct Jacobi-weighted orthogonal polynomials

$\mathcal{P}_{n,r}^{(\alpha, \beta ,\gamma)}(u,v,w), \ \alpha,\beta,\gamma > -1, \ \alpha+\beta+\gamma=0$ , on the triangular domain $T$ . We show that these polynomials $\mathcal{P}_{n,r}^{(\alpha, \beta ,\gamma)}(u,v,w)$ over the triangular domain $T$ satisfy the following properties: $\mathcal{P}_{n,r}^{(\alpha, \beta ,\gamma)}(u,v,w)\in {\mathcal L}_{n}$, $n\geq 1$ , $r=0,1,\dotsc,n,$ and ${\mathcalP}_{n,r}^{(\alpha,\beta,\gamma)}(u,v,w) \perp {\mathcal P}_{n,s}^{(\alpha,\beta,\gamma)}(u,v,w)$ for $r\neq s$ . And hence, ${\mathcalP}_{n,r}^{(\alpha,\beta,\gamma)}(u,v,w)$ , $n=0,1,2,\ldots$ , $r=0,1,\dotsc,n$ form an orthogonal system over the triangular domain $T$ with respect to the Jacobi weight function. These

Jacobi-weighted orthogonal polynomials on triangular domains are

given in Bernstein basis form and thus preserve many properties of

the Bernstein polynomial basis; see the articles on RG:

https://www.researchgate.net/publication/27363224_Jacobi-weighted_orthogonal_polynomials_on_triangular_domains?ev=prf_pub

And

https://www.researchgate.net/publication/220348406_Bivariate_orthogonal_polynomials_on_triangular_domains?ev=prf_pub

Article Jacobi-weighted orthogonal polynomials on triangular domains

Article Bivariate orthogonal polynomials on triangular domains

Dear Dr. Abedallah M Rababah,

thank you very much for your too much cooperation.I found the work very interesting and helpfull.

Dear Khalil,

See the attached paper. I think it will help you.

Dear Parveen Agarwal,

I am very thank full for your cooperation.I found the work very helpfull for my current research.