Applications Of Shuffle Product In Algebra And Number Theory

The shuffle product is an important concept in algebra, combinatorics and topol-ogy. Several generalizations of it have been found in recent years that applied to algebra, combinatorics and number theory. In this paper we study the applications of the shuffle product in algebra and number theory.Rota-Baxter algebra originated in probability theory and then was related to the study of the quantum field and renormalization as a basic structure. Later Rota-Baxter algebra was also combined with the study of associative Yang-Baxter Equation and dendriform algebra. Since the free commutative Rota-Baxter algebra has been described by the way of mixable shuffle product, the study on Rota-Baxter algebra be-came systematic and comprehensive. Moreover, the free noncommutative Rota-Baxter algebra as well as the free dendriform algebra could be regarded as noncommutative shuffle product.The concept of a Nijenhuis operator on a Lie algebra originated from the impor-tant concept of a Nijenhuis tensor that was introduced in the study of pseudo-complex manifolds in the1950s and was related to the concepts of Poisson-Nijenhuis manifolds and the classical Yang-Baxter equation in recent years. The recent theoretic devel-opments of Nijenhuis algebras have largely extended to associative algebra and the construction of free commutative Nijenhuis algebra could be described by another generalization of the shuffle product.In chapter2of this paper. We first give an explicit construction of free Nijenhuis algebras by the way of brackets and then apply it to obtain the universal enveloping Nijenhuis algebra of an NS algebra. We further apply the construction to determine the binary quadratic nonsymmetric algebra, called the N-dendriform algebra, that is compatible with the Nijenhuis algebra. N-dendriform algebra can be regarded as the noncommutative form of the above generalized shuffle product.The main part of the paper is the study on the multiple zeta values(MZVs) by means of shuffle product. Multiple zeta values are the generalizations of the Riemann zeta values at positive integers s1,…,sk. The product of the two MZV not only satisfies the mixable shuffle product of weight1, but also satisfies the mixable shuffle product of weight0. We obtain the algebraic and linear relations between them through the two different double shuffle relation.In chapter3we first give the method of Eic and then get some equations related to Bernoulli numbers and in chapter4we get sixfold restricted sum formula for multiple zeta valves at even integers by the method of recursion that is easier than the way of obtaining the fourfold and fivefold restricted sum formula for multiple zeta valves at even integers. In chapter5we first give some equations related to Bernoulli numbers and then obtain some new families of weighted sum formulas for multiple zeta values according to the relation between Bernoulli numbers and zeta values. In the end of the paper we obtain recursive formulas for the shuffle product and apply it to derive two restricted decomposition formulas for multiple zeta values, which generalize the known results.