Inhomogeneous linear equation in Rota-Baxter algebra

TL;DR

This paper explores solutions to non-homogeneous linear equations within Rota-Baxter algebras, generalizing Spitzer's identity and deriving Eulerian identities through algebraic and combinatorial methods.

Contribution

It introduces a method for solving non-homogeneous equations in Rota-Baxter algebras and extends Spitzer's identity to both commutative and non-commutative cases.

Findings

Generalized Spitzer's identity in Rota-Baxter algebras

Derived Eulerian identities from algebraic structures

Provided explicit solutions for non-homogeneous equations

Abstract

We consider a complete filtered Rota-Baxter algebra of weight $\lambda$ over a commutative ring. Finding the unique solution of a non-homogeneous linear algebraic equation in this algebra, we generalize Spitzer's identity in both commutative and non-commutative cases. As an application, considering the Rota-Baxter algebra of power series in one variable with q-integral as the Rota-Baxter operator, we show certain Eulerian identities.