  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.
