Generating functions from the viewpoint of Rota-Baxter algebras

TL;DR

This paper explores the algebraic structure of generating functions within Rota-Baxter algebras, providing new formulas and applications to classical and novel number sequences.

Contribution

It introduces a framework using complete free Rota-Baxter algebras to generalize generating functions and their formulas, connecting algebraic structures with combinatorial sequences.

Findings

Generalized product and composition formulas for exponential power series

Derived generating functions for classical sequences like Stirling numbers and partition numbers

Introduced generating functions for new combinatorial number families

Abstract

We study generating functions in the context of Rota-Baxter algebras. We show that exponential generating functions can be naturally viewed in a very special case of complete free commutative Rota-Baxter algebras. This allows us to use free Rota-Baxter algebras to give a broad class of algebraic structures in which generalizations of generating functions can be studied. We generalize the product formula and composition formula for exponential power series. We also give generating functions both for known number families such as Stirling numbers of the second kind and partition numbers, and for new number families such as those from not necessarily disjoint partitions and partitions of multisets.