An identity conjectured by Lacasse via the tree function

TL;DR

This paper presents a concise proof of Lacasse's combinatorial identity using the tree function, connecting it to Lambert's W-function and Ramanujan's Q-function, contributing to the understanding of learning theory.

Contribution

It offers a new, shorter proof of Lacasse's conjectured identity by leveraging the properties of the tree function and related special functions.

Findings

Provides a novel, concise proof of Lacasse's identity

Establishes links between the tree function, Lambert's W-function, and Ramanujan's Q-function

Enhances understanding of combinatorial identities in learning theory

Abstract

A. Lacasse conjectured a combinatorial identity in his study of learning theory. Various people found independent proofs. Here is another one that is based on the study of the tree function, with links to Lamberts $W$-function and Ramanujan's $Q$-function. It is particularly short.