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arXiv · 1706.00165

Woon's tree and sums over compositions

C. Vignat·T. Wakhare
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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.

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C. Vignat, T. Wakhare. 2017-06-01. Woon's tree and sums over compositions. https://arxiv.org/abs/1706.00165

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