The Generating Function Method for Quadratic Asymptotically Symmetric Birth and Death Processes

The general form of the generating function associated with quadratic asymptotically symmetric birth and death processes is given with no restriction but the positivity of the transition rates. The explicit form of the moments is obtained when their behaviour is explosive for large time. Finally, the relationship between the generating function method and the Karlin–McGregor approach based on orthogonal polynomial theory is established.