On identities generated by compositions of positive integers

TL;DR

This paper proves surprising identities related to compositions of positive integers, introduces new identities for Stirling numbers of the first kind, and explores their algebraic structures and related identities.

Contribution

It presents novel identities for compositions and Stirling numbers, along with an algebraic framework and structural insights not previously documented.

Findings

New identities for Stirling numbers of the first kind

Structural algebraic interpretations of the identities

Additional related identities derived from the main results

Abstract

We prove astonishing identities generated by compositions of positive integers. In passing, we obtain two new identities for Stirling numbers of the first kind. In the two last sections we clarify an algebraic sense of these identities and obtain several other structural close identities.