Unimolecular reaction kinetics

We use generating functions to analyze the kinetics of the reaction $A+B\to (1+\epsilon )A+B$ where $-1\mathrm{}<~\epsilon <0\mathrm{}$ and $\epsilon >0\mathrm{}$ correspond, respectively, to $B$ particles that are traps or "sources." For an arbitrary configuration and strengths of static $B$'s and mobile $A$'s, an exact formal expression for the kinetics of the reaction is derived. This solution yields either exponential growth, power-law growth or decay, or no growth asymptotically, depending on the configuration and strength of the $B$'s.