Negative binomial sums of random variables and discounted reward processes

Given a sequence of random variables (rewards), the Haviv-Puterman differential equation relates the expected infinite-horizon lambda-discounted reward and the expected total reward up to a random time that is determined by an independent negative binomial random variable with parameters 2 and lambda. This paper provides an interpretation of this proven, but previously unexplained, result. Furthermore, the interpretation is formalized into a new proof, which then yields new results for the general case where the rewards are accumulated up to a time determined by an independent negative binomial random variable with parameters k and lambda.