Stieltjes functions and discrete classical orthogonal polynomials

Classical orthogonal polynomials can be characterized in terms of the corresponding Stieltjes function. We consider the construction of a Stieltjes function in terms of the falling factorials for discrete classical orthogonal polynomials (Charlier, Krawtchouk, Meixner, and Hahn). This Stieltjes function associated with classical orthogonal polynomials of a discrete variable is solution of a non-homogeneous difference equation. That property characterizes the discrete classical measures. In addition, an hypergeometric expression for the Stieltjes function is obtained in all the discrete classical cases.