In this paper we obtain a set of polynomials which are orthogonal with respect to the classical discrete weight function of the Charlier polynomials at which an extra point mass at x = 0 is added. We construct a difference operator of infinite order for which these new discrete orthogonal polynomials are eigenfunctions.

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Abstract. The Sobolev-type Laguerre polynomials {L α,M,N n (x)} ∞ n=0 are orthogonal with respect to the inner product,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 formwhereare independent of n. This differential operator is of order 2α + 4 if α is a nonnegative integer, and of infinite order otherwise.In this paper we construct all differential equations of the formwhere the coefficientsare 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 α,M,N n (x)} ∞ n=0 . 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 2α + 8 if α 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 4α + 10 if α is a nonnegative integer and of infinite order otherwise.

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