The Cummings-Stell model of associative fluids: a general solution

In a series of publications the Cummings-Stell model (CSM), for a binary mixture of associative fluids with steric effects, has been solved analytically using the Percus-Yevick approximation (PYA). The solution consists in a square well potential of width w, whose center is placed into the hard sphere shell ($r < \sigma$): at $L = \sigma / n$ (n = 1, ..., 4). This paper presents a general solution, for any n, of the first order Difference Differential Equation (DDE), for the auxiliary Baxter's function that appears in the CSM, using recursive properties of these auxiliary functions and a matrix composed by differential and shift operators (MDSO). This problem is common in some other models of associative fluids such as the CSM for homogeneus and inhomogeneus mixtures of sticky shielded hard spheres including solvent effects under PYA, and in that of mean-spherical approximation (MSA), for chemical ion association and dipolar dumbbells and polymers. The sticky potential implies a discontinuity step at $L$ in the solution of auxiliary Baxter's functions so that, one side, $L$ now is arbitrary and, for some additional effects, it can be placed one or more sticky potentials at different positions into the hard shell.