# Woon's tree and sums over compositions

## TL;DR

This paper explores sums over compositions of integers, deriving generating functions, connecting to recursive trees, and extending these concepts to nonlinear transforms and generalized sums, with applications to special functions like Bernoulli numbers.

## Contribution

It introduces a new framework linking composition sums with recursive trees and nonlinear transforms, extending Fuchs' general PI tree to generalized sums over compositions.

## Key findings

- Derived a generating function for sums over compositions.
- Connected composition sums with recursive trees and nonlinear transforms.
- Extended the concept to generalized sums over parts of compositions.

## Abstract

This article studies sums over all compositions of an integer. We derive a generating function for this quantity, and apply it to several special functions, including various generalized Bernoulli numbers. We connect composition sums with a recursive tree introduced by S.G. Woon and extended by P. Fuchs under the name "general PI tree", in which an output sequence $\{x_n\}$ is associated to the input sequence $\{g_n\}$ by summing over each row of the tree built from $\{g_n\}$. Our link with the notion of compositions allows to introduce a modification of Fuchs' tree that takes into account nonlinear transforms of the generating function of the input sequence. We also introduce the notion of \textit{generalized sums over compositions}, where we look at composition sums over each part of a composition.