Self-adjoint differential equations for classical orthogonal polynomials

This paper deals with spectral type differential equations of the self-adjoint differential operator, 2 r order: L ( 2 r ) [ Y ] ( x ) = 1 ρ ( x ) d r d x r ρ ( x ) β r ( x ) d r Y ( x ) d x r = λ rn Y ( x ) . If ρ ( x ) is the weight function and β ( x ) is a second degree polynomial function, then the corresponding classical orthogonal polynomials, { Q n ( x ) } n = 0 ∞ , are shown to satisfy this differential equation when λ rn is given by λ rn = ∏ k = 0 r - 1 ( n - k ) [ α 1 + ( n + k + 1 ) β 2 ] , where α 1 and β 2 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 Q n ( x ) and λ rn as eigenfunctions and eigenvalues, respectively. Any linear combination of such self-adjoint operators has Q n ( x ) as eigenfunctions and the corresponding linear combination of λ rn as eigenvalues.