A new integral-series identity of multiple zeta values and regularizations (Q1652901)

This paper gives an identity, called the integral-series identity, between multiple zeta values. The identity is magnificent in that despite being `very simple and elementary' (in the authors' words), it conjectually yields all linear relations among multiple zeta values. The paper gives a compelling evidence of this conjecture by showing that the identity is equivalent to the regularization theorem established by Ihara, Kaneko, and Zagier [\textit{K. Ihara} et al., Compos. Math. 142, No. 2, 307--338 (2006; Zbl 1186.11053)]. The sections are structured as follows. Section 1 is the introduction. Section 2 introduces notation and the algebraic setup to be used throughout the paper. Section 3 reviews what is sometimes known as the Yamamoto integral of multiple zeta(-star) values. Section 4 states the main theorems: the integral-series identity itself, and the equivalence of the integral-series identity, the regularization theorem, and the star-regularization theorem. Section 5 gives proofs of the main theorems described in Section 4. Section 6 discusses the relationship between the integral-series identity and Kawashima's relation, and proves that the double shuffle relation, the regularization theorem (or equivalently the integral-series identity), and the duality relation together imply Kawashima's relation. Section 7 shows that the integral-series identity implies the restricted sum formula (and so the sum formula as well) and Hoffman's relation.