The tennis ball problem

Our object is to explore "the s-tennis ball problem" (at each turn s balls are available and we play with one ball at a time). This is a natural generalization of the case s=2 considered by Mallows and Shapiro. We show how this generalization is connected with s-ary trees and employ the notion of generating trees to obtain a solution expressed in terms of generating functions. Then, we present a particular case in which at each turn we have 4 balls and play with 2 balls at a time. To solve this problem we use the concepts of Riordan Arrays and stretched Riordan Arrays, and a generalization of generating trees. This is a joint work with D. G. Rogers, R. Sprugnoli and M. C. Verri.