Partial differential equations and bivariate orthogonal polynomials
The author solves the Non-Hermitian Bochner Problem (NHBP): find all triples \((L_1, L_2, P)\) where \(P\) is a family of bivariate orthogonal polynomials and \(L_1\) and \(L_2\) are the second-order linear partial differential operators which satisfy such conditions \(\phi (1)=1,\) \(\phi (\overline{z})=\overline{\phi (z)},\) \(\overline{\phi}\in P,\) \(z\mapsto z\) belongs to \(P,\) \(L_1(\phi)=\phi _x(1)\phi\) and \(L_2(\phi)=\phi_y(1)\phi,\) \(\phi \in P\) (\(\phi_x\) and \(\phi_y\) are the partial derivatives). One of the main results is: Theorem. The only solutions to the NHBP are \((L_1^\gamma, L_2^\gamma, D^\gamma)\) for \(\gamma >-1.\) Here are \(L_1^\gamma \phi =x\phi _x+ y\phi _y +\frac {1}{2\gamma +2}(x^2+y^2-1)(\phi _{xx}+ \phi {yy}),\) \(L_2^\gamma \phi = x\phi _y- y\phi _x,\) \(D^\gamma= \{R_{m,n}^\gamma (x,y): m,n \in N_0\},\) \(R_{m,n}^\gamma (x,y)= r^{|m-n|}e^{i(m-n)\theta } R_{m\wedge n}^{\gamma ,|m-n|} (2r^2-1),\) \(r=|x+iy|,\) \(\theta =\text{arg} (x+iy),\) \(R_{n}^{\alpha ,\beta }\) the normalized Jacobi polynomials \((R_{n}^{\alpha ,\beta }(1)=1).\) The author uses Mathematica in the proof.
second-order linear partial differential operators
scientific article; zbMATH DE number 1423193