Nested Recursions, Simultaneous Parameters and Tree Superpositions

Abstract

We apply a tree-based methodology to solve new, very broadly defined families of nested recursions of the general form $R(n)=\sum_{t=1}^k R(n-a_t-\sum_{i=1}^{p}R(n-b_{ti}))$, where $a_t$ are integers, $b_{ti}$ are natural numbers, and $k,p$ are natural numbers that we use to denote "arity" and "order," respectively, and with some specified initial conditions. The key idea of the tree-based solution method is to associate such recursions with infinite labelled trees in a natural way so that the solution to the recursions solves a counting question relating to the corresponding trees. We characterize certain recursion families within $R(n)$ by introducing "simultaneous parameters" that appear both within the recursion itself and that also specify structural properties of the corresponding tree. First, we extend and unify recently discovered results concerning two families of arity $k=2$, order $p=1$ recursions. Next, we investigate the solution of nested recursion families by taking linear combinations of solution sequence frequencies for simpler nested recursions, which correspond to superpositions of the associated trees; this leads us to identify and solve two new recursion families for arity $k=2$ and general order $p$. Finally, we extend these results to general arity $k>2$. We conclude with several related open problems.