Exact solutions of some quadratic and quartic birth and death processes and related orthogonal polynomials

We study a birth and death process with quartic transition rates for which the transition probabilities (but for the honest ones) are not unique. For their related polynomials we exhibit infinitely many different orthogonality measures among which stems the honest one. These results are obtained firstly by reducing the Kolmogorov equations to linear partial differential equations and secondly by mapping them to partial differential equations with constant coefficients which can be solved analytically.