Combinatorial Identities from the Generating Functions of Two Binomial Coefficients

• Main Authors: Shing-Jing Tsai, 蔡幸靜
• Other Authors: Minking Eie
• Format: Others
• Language: en_US
• Published: 2014
• Online Access: http://ndltd.ncl.edu.tw/handle/7uh3gp

碩士 === 國立中正大學 === 數學研究所 === 102 === In recent years, it is interesting to know more about the multiple zeta values. In this thesis, we shall take a look at some histories about the multiple zeta values in Section 1. In Section 2, we focus on the Drinfel’d integral and its applications. Drinfel’d integral is a very useful representation for a multiple zeta value, so that we can express a multiple zeta value as an integral of a string of differential forms of two simple types instead of summation forms. In particular, the number of variables in the integral can be reduced. Also, there are many ways to express a multiple zeta value of height one. Nevertheless, each of the expressional forms has its own specific viewpoint. In Section 3, we concentrate our attention on the combinatorial identities obtained from shuffle relations. By counting the number of the multiple zeta values, we also obtain new combinatorial identities. Especially interesting are those integrals obtained from shuffle products of two sets of multiple zeta values; the resulting combinatorial identities are very interesting but hard to prove as usual. In Section 4, we shall discuss some specific combinatorial identities from the generating functions of two binomial coefficients. More interestingly, we found that if we focus on the same multiple zeta value ζ(j+β−i+2) and ζ(α−j+i+2) (for integers α,β,i,j with 0 ≤ j ≤ α and 0 ≤ i ≤ β), but consider two different integral expressions, then we get two combinatorial identities from the generating functions of two binomial coefficients among the two multiple zeta values, which are our Main Theorems A and B.