Centrally symmetric orthogonal polynomials and second order partial differential equations (Q5937111)

Orthogonal polynomial solutions to a partial differential equation of the form NEWLINE\[NEWLINE Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}=\lambda_nu\tag{*} NEWLINE\]NEWLINE are considered. Equation (*) has a polynomial system \(\{\Phi_n\}_{n=0}^\infty\) as solutions if \(A=ax^2+d_1x+e_1y+f_1\), \(2B=2axy+d_2x+e_2y+f_2\), \(C=ay^2+d_3x+e_3y+f_3\), \(D=gx+h_1\), \(E=gy+h_2\), \(\lambda_n=na(n-1)+gn\) [see \textit{H. L. Krall} and \textit{I. M. Sheffer}, Ann. Mat. Pura Appl. (4) 76, 325-376 (1967; Zbl 0186.38602)]. Under these conditions on the coefficients of (*) it is proved that (*) has a centrally symmetric (\(\Phi_n(-x,-y)=(-1)^n \Phi_n(x,y)\)) orthogonal polynomial system of solutions if and only if \(d_1=e_1=d_2=e_2=d_3=e_3=h_1=h_2=0\) and \(a\not=g\), \(\Delta\equiv f_2^2- 4f_1f_3\not=0\). Moreover, by a suitable real linear change of variables every equation of the form NEWLINE\[NEWLINE(ax^2+f_1)u_{xx}+(2axy+f_2)u_{xy}+(ay^2+f_3)u_{yy} +g(xu_{x}+yu_{y})=\lambda_nuNEWLINE\]NEWLINE can be transformed into one of seven equations. For example, if \(\Delta<0\), \(a=0\), \(gf_1>0\) this equation has the form NEWLINE\[NEWLINEu_{xx}+u_{yy}+2xu_{x}+2yu_{y}=2nu.NEWLINE\]