Symmetric Jacobi Polynomials on a Triangle and Their Spectral Algebra

We study a family of symmetric orthogonal polynomials on the unit triangle associated with the weight $$w_{\alpha,\gamma,\kappa}(x,y) = (xy)^\alpha(1-x-y)^\gamma |x-y|^{2\kappa+1}, \quad \alpha,\gamma,\kappa>-1,\quad \alpha+\kappa>-\frac32.$$ We construct the corresponding monic symmetric orthogonal basis on the simplex chamber and prove that its elements are eigenfunctions of a formally self-adjoint second-order differential operator $\mathcal D_1^{\alpha,\gamma,\kappa}$. A pair of adjoint ladder operators yields a second operator of order two $\mathcal D_2^{\alpha,\kappa}$ with the same eigenfunctions. We also obtain an explicit representation of the basis in terms of one-variable Jacobi polynomials and compute its squared norms. After passing to the elementary symmetric variables, we determine the full algebra of linear partial differential operators with real polynomial coefficients having the transformed polynomials as eigenfunctions. Every such operator can be written uniquely as a polynomial in $\mathcal D_1^{\alpha,\gamma,\kappa}$ and $\mathcal D_2^{\alpha,\kappa}$; consequently, this algebra is isomorphic to the real polynomial ring in two variables.