Karlin-McGregor-like formula in a simple time-inhomogeneous birth-death   process
  A novel approach is employed and developed to derive transition probabilities for a simple time-inhomogeneous birth-death process. Algebraic probability theory and Lie algebraic treatments make it easy to treat the time-inhomogeneous cases. As a result, an expression based on the Charlier polynomials is obtained, which can be considered as an extension of a famous Karlin-KcGregor representation for a time-homogeneous birth-death process. 
Jun Ohkubo
math-ph cond-mat.stat-mech math.MP