Contents -- Preface -- I Basic Theory of Multiple Zeta Values -- 0 The Time Before Multiple Zeta Values -- 0.1 The Evaluation of Euler Double Sums -- 0.2 Vandermonde Convolution -- 0.3 Zeta Functions Associated with Multiple Zeta Values -- 0.4 Messages from Modular Forms -- 1 Introduction to the Theory of Multiple Zeta Values -- 1.1 Introduction and Notations -- 1.2 Drinfeld Integral Representations of Multiple Zeta Values -- 1.3 Double Weighted Sum Formulas -- 1.4 The Expectations of Binomial Distributions -- 1.5 Exercises -- 2 The Sum Formula -- 2.1 Through the Integral Representations -- 2.2 Another Proof of the Sum Formula -- 2.3 Evaluation of Multiple Zeta Values of Height One -- 2.4 Exercises -- II Shuffle Relations among Multiple Zeta Values -- 3 Some Shuffle Relations -- 3.1 Shuffle Relations of Multiple Zeta Values -- 3.2 An Application of Double Weighted Sums -- 3.3 Shuffle Relations of Two Sums of Multiple Zeta Values -- 3.4 A Vector Version of the Restricted Sum Formula -- 3.5 Exercises -- 4 Euler Decomposition Theorem -- 4.1 A Shuffle Relation with Two Parameters -- 4.2 Integrals with Three Factors -- 4.3 Generalizations of Euler Decomposition Theorem -- 4.4 Applications of the Decomposition Theorem -- 4.5 Applications of Another Decomposition Theorem -- 4.6 Exercises -- 5 Multiple Zeta Values of Height Two -- 5.1 Sums of Multiple Zeta Values of Height Two -- 5.2 Weighted Sums of Multiple Zeta Values of Height Two -- 5.3 The Shuffle Product Formula of a Sum and Others -- 5.4 Exercises -- III Applications of Shuffle Relations in Combinatorics -- 6 Generalizations of Pascal Identity -- 6.1 Applications of Shuffle Products in Combinatorics -- 6.2 Hypergeometric Distribution -- 6.3 The Generating Function of Three Variables -- 6.4 Exercises -- 7 Combinatorial Identities of Convolution Type -- 7.1 Some Particular Combinatorial Identities.

7.2 A Generating Function for Products -- 7.3 A Combinatorial Identity of Convolution Type -- 7.4 Another Generating Function of Three Variables -- 7.5 Exercises -- 8 Vector Versions of Some Combinatorial Identities -- 8.1 The Shuffle Product of Two Sums -- 8.2 More Combinatorial Identities of Convolution Type -- 8.3 Vector Versions of Pascal Identity -- 8.4 Problems on Combinatorial Identity -- Appendices -- A Singular Modular Forms on the Exceptional Domain -- A.1 Cayley Numbers and Integral Cayley Numbers -- A.2 The Exceptional Domain -- A.3 The Theory of Jacobi Forms -- A.4 A Final Application -- Appendix (i): Jacobi Forms over Cayley Numbers -- Appendix (ii): Basic Properties of a Set of Theta Series -- B Shuffle Product Formulas of Multiple Zeta Values -- B.1 Introduction -- B.2 The Shuffle Product Formula of Two Multiple Zeta Values -- B.3 Some Basic Shuffle Relations -- B.4 Shuffle Relations of Two Sums of Multiple Zeta Values -- B.5 The Generating Function of Height One -- Appendix (i): Double Weighted Sum Formulas -- Appendix (ii): Evaluations of Some Particular Integrals -- C The Sum Formula and Their Generalizations -- C.1 The Double Integral Representation of a Sum -- C.2 Euler Sums with Two Branches -- C.3 Another Application of Euler Sums with Two Branches -- C.4 The Restricted Sum Formula -- C.5 A Generating Function for Sums of Multiple Zeta Values -- C.6 The Vector Version of the Restricted Sum Formula -- C.7 Another Generating Function -- Bibliography -- Index.