The authors study the properties of the generalized Meixner polynomials \(M^{\gamma, \mu, A}_n(x)\). These ones are orthogonal with respect to the linear functional \(U\) on the space of polynomials with real coefficients, \[ \langle U, P\rangle= \sum_{x\in \mathbb{N}} {\mu^x \Gamma(\gamma+ x)\over \Gamma(\gamma) \Gamma(1+ x)} P(x)+ AP(0),\quad x\in\mathbb{N},\quad A\geq 0. \] An expression of \(M^{\gamma, \mu, A}_n(x)\) in terms of classical Meixner polynomials \(M^{\gamma, \mu, 0}_n(x)\) and its first difference derivative is found. This expression is used to establish a representation of \(M^{\gamma, \mu, A}_n(x)\) as a hypergeometric function \(_3F_2\) and to obtain the second-order difference equation for the generalized Meixner polynomials. This gives the solution of a problem posed by \textit{R. Askey} [``Orthogonal polynomials and their applications'', p. 418, Ann. Comput. Appl. Math. 9, Baltzer AG Scientific, Basel (1991)]. (See also another approach to R. Askey's problem given by \textit{H. Bavinck} and \textit{H. van Haeringen}, J. Math. Anal. Appl. 184, No. 3, 453-463 (1994; Zbl 0824.33005)).