Fourth-order difference equation for co-recursive associated Meixner and Charlier polynomials

The families of Meixner and Charlier polynomials play an important part in the solution of the Chapman–Kolmogorov equation of linear birth and death processes. We present an explicit representation of the co-recursive associated Meixner polynomials in terms of hypergeometric functions. This representation allows to derive the fourth-order difference equation satisfied by these polynomials, a generating function and the Stieltjes transform of the orthogonality measure. Special attention is given on certain simple limiting cases occurring in the solutions of the equations of linear birth and death processes.