On differential equations for Sobolev-type Laguerre polynomials

The Sobolev-type Laguerre polynomials {L-n(alpha,M,N)(x)}(n=0)(infinity) are orthogonal with respect to the inner product [f,g] = 1/Gamma(alpha + 1) integral(0)(infinity) x(alpha)e(-x)f(x)g(x)dx + Mf(0)g(0) + Nf'(0)g'(0), where alpha > -1, M equal to or greater than 0 and N equal to or greater than 0. In 1990 the first and second author showed that in the case M > 0 and N = 0 the polynomials are eigenfunctions of a unique differential operator of the form M Sigma(i=1)(infinity)a(i)(x)D-i + xD(2) + (alpha + 1 -x)D, where {a(i)(x)}(i=1)(infinity) are independent of n. This differential operator is of order 2 alpha + 4 if a is a nonnegative integer, and of infinite order otherwise. In this paper we construct all differential equations of the form M Sigma(i=0)(infinity) a(i)(x)y((i))(x) + N Sigma(i=0)infinity b(i)(x)y((i))(x) + MN Sigma(i=0)(infinity) c(i)(x)y((i))(x) + xy"(x) + (alpha + 1 - x)y'(x) + ny(x) = 0, where the coefficients {a(i)(x)}(i=1)(infinity), {b(i)(x)}(i=1)(infinity) and {c(i)(x)}(i=1)infinity are independent of n and the coefficients a(0)(x), b(0)(x) and c(0)(x) are independent of x, satisfied by the Sobolev-type Laguerre polynomials {L-n(alpha,M,N)(x)}(n=0)(infinity). Further, we show that in the case M = 0 and N > 0 the polynomials are eigenfunctions of a linear differential operator, which is of order 2 alpha + 8 if a is a nonnegative integer and of infinite order otherwise. Finally, we show that in the case M > 0 and N > 0 the polynomials are eigenfunctions of a linear differential operator, which is of order 4 alpha + 10 if alpha is a nonnegative integer and of infinite order otherwise.