On the fourth order differential equation having orthogonal polynomials solutions

Abstract

Necessary and sufficient conditions for an orthogonal polynomial system (OPS) to satisfy a differential equation with polynomial coefficients of form $$L_N[y]=\sum^N_{i=1}\ell_i(x)y^{(i)}(x)=\lambda_ny(x)$$ were found by H.L.Krall and he Classified orthogonal polynomials satisfying fourth order differential equation. Here, we find a necessary condition for the above differential equation with N=4 to have an OPS as solution and using the new condition, we classify all orthogonal polynomials satisfying the fourth order differential equation up to real change of variable.