Differential equations of infinite order for Sobolev-type orthogonal polynomials

Abstract

Assume that {P-n(x)}(infinity)(n=0) are orthogonal polynomials relative to a quasi-definite moment functional sigma, which satisfy a differential equation of spectral type of order D (2 less than or equal to D less than or equal to infinity): L(D)[y](x) = (i=1)Sigma(D) li(x)y((i))(x) = lambda(n)y(x), where l(i)(x) are polynomials of degree less than or equal to i. Let phi be the symmetric bilinear form of discrete Sobolev type defined by phi(p,q) = (sigma,pq) + Np-(k)(c)q((k))(c), where N(not equal 0) and c are real constants, k is a non-negative integer, and p and q are polynomials. We first give a necessary and sufficient condition for phi to be quasi-definite and then show: If phi is quasi-definite, then the corresponding Sobolev-type orthogonal polynomials {R(n)(N,k;c)(x)}(infinity)(n=0) satisfy a differential equation of infinite order of the form N{a(0)(x,n)y(x) + (i=1)Sigma(infinity)a(i)(x)y((i))(x)} + L(D)[y](x) = lambda(n)y(x), where {a(i)(x)}(infinity)(i=0) are polynomials of degree less than or equal to i, independent of n except a(0)(x) := a(0)(x,n). We also discuss conditions under which such a differential equation is of finite order when a is positive-definite, D < infinity, N greater than or equal to 0, and k = 0.