Self-adjoint differential equations for classical orthogonal polynomials

This paper deals with spectral type differential equations of the self-adjoint differential operator, 2r order:L(2r)[Y](x)=1ρ(x)drdxrρ(x)βr(x)drY(x)dxr=λrnY(x).If ρ(x) is the weight function and β(x) is a second degree polynomial function, then the corresponding classical orthogonal polynomials, {Qn(x)}n=0∞, are shown to satisfy this differential equation when λrn is given byλrn=∏k=0r-1(n-k)[α1+(n+k+1)β2],where α1 and β2 are the leading coefficients of the two polynomial functions associated with the classical orthogonal polynomials. Moreover, the singular eigenvalue problem associated with this differential equation is shown to have Qn(x) and λrn as eigenfunctions and eigenvalues, respectively. Any linear combination of such self-adjoint operators has Qn(x) as eigenfunctions and the corresponding linear combination of λrn as eigenvalues.