Explicit Relations between Kaneko--Yamamoto Type Multiple Zeta Values and Related Variants

Abstract

In this paper we first establish several integral identities. These integrals are of the form \[\int_0^1 x^{an+b} f(x)\,dx\quad (a\in\{1,2\},\ b\in\{-1,-2\})\] where $f(x)$ is a single-variable multiple polylogarithm function or $r$-variable multiple polylogarithm function or Kaneko--Tsumura A-function (this is a single-variable multiple polylogarithm function of level two). We find that these integrals can be expressed in terms of multiple zeta (star) values and their related variants (multiple $t$-values, multiple $T$-values, multiple $S$-values etc.), and multiple harmonic (star) sums and their related variants (multiple $T$-harmonic sums, multiple $S$-harmonic sums etc.). Using these integral identities, we prove many explicit evaluations of Kaneko--Yamamoto multiple zeta values and their related variants. Further, we derive some relations involving multiple zeta (star) values and their related variants.

31 pages, section 1 revised to add connections to the Schur multiple zeta values

References in corpus (3)

• Multiple zeta-star values and multiple integrals
• Variants of Multiple Zeta Values with Even and Odd Summation Indices
• Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series

Cited by in corpus (4)

• Apéry-Type Series and Colored Multiple Zeta Values
• Weighted and Restricted Sum Formulas of Euler Sums
• Explicit Relations of Some Variants of Convoluted Multiple Zeta Values
• Explicit Evaluation of Euler-Apéry Type Multiple Zeta Star Values and Multiple $t$-Star Values