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)= 1 (x) d dxr ( (x) r (x) dY (x) dxr ) = rnY (x). If (x) is the weight function and (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 rn is given by rn = r−1 ∏ k=0 (n− k)[ 1 + (n+ k + 1) 2], where 1 and 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 Qn(x) and rn as eigenfunctions and eigenvalues, respectively.Any linear combination of such self-adjoint operators hasQn(x) as eigenfunctions and the corresponding linear combination of rn as eigenvalues. © 2004 Elsevier B.V. All rights reserved.