Integrals of logarithmic functions and alternating multiple zeta values

Abstract

By using the method of iterated integral representations of series, we establish some explicit relationships between multiple zeta values and Integrals of logarithmic functions. As applications of these relations, we show that multiple zeta values of the formfor p = 1 and 2, satisfy certain recurrence relations which allow us to write them in terms of zeta values, polylogarithms and ln 2. Moreover, we also prove that the multiple zeta values ζ(1, {1} m−1 , 3, {1} k−1 ) can be expressed as a rational linear combination of products of zeta values, multiple polylogarithms and ln 2 when m = k ∈ N. Furthermore, we also obtain reductions for certain multiple polylogarithmic values at 1 2 .

Evaluations of Euler-Type Sums of Weight $$\le 5$$≤5

“…In this section, we give some lemmas which will be useful in the development of the main results. 25]) For integers m ≥ 1 and k ≥ 0, then the following identity holds:…”

Linear relations between logarithmic integrals of high weight and some closed-form evaluations

“…(See Section 4.1). An expression for ia0c2 in terms of MZVs and multiple polylogarithm is given by Xu at [5], but it seems not directly applicable for our problem.…”

Identities for the Multiple Zeta (Star) Values

“…In [23], we considered the following restricted sum formulas involving multiple zeta values ln k (1 + t m+p+2 ) (1 + t 1 ) · · · (1 + t m ) t m+1 · · · t m+p+1 t m+p+2 dt 1 · · · dt m+p+2 .…”