  The Rota-Baxter operator and the modified Rota-Baxter operator on various
algebras are both important in mathematics and mathematical physics. The former
is originated from the integration-by-parts formula and probability with
applications to the renormalization of quantum field theory and the classical
Yang-Baxter equation. The latter originated from Hilbert transformations with
applications to ergodic theory and the modified Yang-Baxter equation. Their
merged form, called the extended Rota-Baxter operators, has also found
interesting applications recently.
  This paper presents a systematic study of the extended Rota-Baxter operator.
We show that while extended Rota-Baxter operators have properties similar to
Rota-Baxter operators; they provide a linear structure that unifies Rota-Baxter
operators and modified Rota-Baxter operators. Examples of extended Rota-Baxter
operators are also given, especially from polynomials and Laurent series due to
their importance in ($q$-)integration and the renormalization in quantum field
theory. We then construct free commutative extended Rota-Baxter operators by a
generalization of the quasi-shuffle product. The multiplication of the initial
object in the category of commutative extended Rota-Baxter operators allows a
combinatorial interpretation in terms of a color-enrichment of Delannoy paths.
Applying its universal property, we equip a free commutative extended
Rota-Baxter operators with a coproduct which has a cocycle condition, yielding
a bialgebraic structure. We then show that this bialgebra on a free extended
Rota-Baxter operators possesses an increasing filtration and a connectedness
property, culminating at a Hopf algebraic structure on a free commutative
extended Rota-Baxter operator.
