Connection problems for polynomial solutions of nonhomogeneous differential and difference equations
The authors consider differential operators of the type \[ L[z(x)]= (s(x)D^2+ t(x)D+\lambda I)[z(x)], \] where \(s\) and \(t\) are polynomials, \(\lambda\) is a constant, \(D= d/dx\), as well as the corresponding discretized versions of such operators. They investigate the behavior of the solutions of differential resp. difference equations \[ L[z(x)]= Q_n(x), \] where \(Q_n\) is a polynomial. Expansions of the solutions with respect to the set of polynomials \(\{Q_n\}\) and a recurrence relation for the coefficients of this expansion are given. Some examples as well as extensions are also provided.
scientific article; zbMATH DE number 1326366