In this paper we first establish several integral identities involving the multiple polylogarithm functions and Kaneko-Tsumura A-function, which can be thought as a single-variable function of level two. We find that these integrals expressed in terms zeta (star) values, their related variants (multiple t-values, T-values, S-values, etc.), harmonic sums T-harmonic sums, S-harmonic are closely t...

In this paper we show how to express the values of fF(3) for arbitrary number field F in terms of the trilogarithms (D. Zagier's conjecture) and how to relate this result to algebraic K-theory. 1. THE CLASSICAL POLYLOGARITHM FUNCTION The classical polylogarithm function n (1.1) Up(z) := ^ ^ ( 2 € C , | 2 | < l , p € N ) n-\ during the last 200 years was the subject of much research—see [L]. Usi...

Let k[ε]2 := k[ε]/(ε2). The single valued real analytic n-polylogarithm Ln : C → R is fundamental in the study of weight n motivic cohomology over a field k, of characteristic 0. In this paper, we extend the construction in Ünver (Algebra Number Theory 3:1–34, 2009) to define additive n-polylogarithms lin:k[ε]2 → k and prove that they satisfy functional equations analogous to those of Ln . Unde...

A new three-dimensional integral containing f(x,y,z)Iv(x?) is derived where Iv(x?) the Modified Bessel Function of first kind and taken over infinite cubic space 0&lt;x&lt;?,0&lt;y&lt;?,0&lt;z&lt;?. The not easily evaluated for complex ranges parameters. representation in terms Hurwitz–Lerch zeta function, polylogarithm function Riemann functions are evaluated. This yields triple representation...

In this paper, we construct a new Eisenstein cocycle called the Shintani-Barnes which specializes in uniform way to values of zeta functions general number fields at positive integers. Our basic strategy is generalize construction presented work Vlasenko and Zagier by using some recent techniques developed Bannai, Hagihara, Yamada, Yamamoto their study polylogarithm for totally real fields. We ...