Differential equations of infinite order for Sobolev-type orthogonal polynomials

Assume that Pn(x)n=0∞ are orthogonal polynomials relative to a quasi-definite moment functional σ, which satisfy a differential equation of spectral type of order D (2⩽D⩽∞)where li(x) are polynomials of degree ⩽i. Let be the symmetric bilinear form of discrete Sobolev type defined by (p,q) = 〈σ, pq〉 + Np(k)(c)q(k)(c) where N(≠0) and c are real constants, k is a non-negative integer, and p and q are polynomials. st give a necessary and sufficient condition for to be quasi-definite and then show: If is quasi-definite, then the corresponding Sobolev-type orthogonal polynomials RnN,k;c(x)n=0∞ satisfy a differential equation of infinite order of the form φ(p,q)+Np(k)(c)q(k)(c)where ai(x)i=0∞ are polynomials of degree ⩽i, independent of n except a0(x) := a0(x,n). We also discuss conditions 3nder which such a differential equation is of finite order when σ is positive-definite, D<∞, N⩾0, and k=0.