Symmetrizability of differential equations having orthogonal polynomial solutions (Q1372111)

The authors reconsider the linear differential equations of spectral type with polynomial coefficients of the form \[ L_N[y](x) =\sum^N_{i=0} l_i(x)y^{(i)} =\lambda_ny(x), \tag{*} \] where \[ l_i(x)= \sum^i_{j=0} l_{ij} x^j, \quad i=0,1, \dots,N \] stands for a polynomial of degree not greater than \(i\) and \(\lambda_n\) is the eigenvalue parameter given by \[ \lambda_n =l_{00} +l_{11} n+\cdots +l_{NN} n(n-1) \dots(n-N+1), \quad n=0,1, \dots \] Their main result is that if the differential equation (*) has an orthogonal polynomial system of solutions, then the differential operator \(L_N[\cdot]\) appearing there is always symmetrizable. Notice that the symmetrizability of \(L_N\) is important when one wants to develop a spectral theory of (*), which seeks self-adjoint differential operators in some Hilbert or Krein space.