In  Karlin and McGregor analyzed a class of linear growth birth and death processes admitting a representation formula of the transition probability function in terms of classical orthogonal polynomials. Birth and death processes belong to the category of reversible stochastic processes (see  and ). In this case by invoking the spectral resolution of the identity for Hermitian operators one achieves the representation formula of Karlin and McGregor described in  and . This theory is mostly restricted to the case of one-dimensional birth and death processes and diffusion processes. Some special higher dimensional birth and death processes motivated by certain applications in studies of population growth are reversible. For these processes there are representation formulae for the transition probability function that can be explicitly determined. In  and  Karlin and McGregor develop a discretized version of the classical technique of solving Laplace's equation in terms of spherical harmonics. This paper applies their method to determine the representation formula for the transition probability function of two two-dimensional (Sections 2 and 3) and the corresponding higher dimensional (Sections 4 and 5) linear growth birth and death processes. In each case the representation formula is expressed explicitly in terms of semi-direct products of classical orthogonal polynomials somewhat reminiscent of such products of spherical harmonics. In the Appendix the properties of the orthogonal polynomials are summarized and the method of derivation of the explicit form of the representation formula is outlined.
"A Multi-dimensional Linear Growth Birth and Death Process." Ann. Math. Statist. 39 (3) 727 - 754, June, 1968. https://doi.org/10.1214/aoms/1177698308