Ramified and Unramified Motivic Multiple tt-, TT- and SS-Values

In this paper we shall consider a few variants of the motivic multiple zeta values of level two by restricting the summation indices in the definition of multiple zeta values to some fixed parity patterns. These include Hoffman's multiple $t$-values, Kaneko and Tsumura's multiple $T$-values, and the multiple $S$-values studied previously by the authors. By applying Brown and Glanois's descent theory on the motivic versions of these values we shall derive some criterion for when these values are ramified and unramified. Assuming Grothendieck's period conjecture, our results partially confirms a conjecture of Kaneko and Tsumura about when multiple $T$-values can be expressed as a rational linear combination of multiple zeta values (i.e., unramified) if their depth is less than four. Similar results are obtained for motivic multiple $S$-values. Further, we are able to generalize a result of Charlton to more families of unramified multiple $t$-values with unit components (i.e. components equal to 1). We propose some more unsolved problems at the end of the paper.Comment: 40 pages, comments welcom

On Some Unramified Families of Motivic Euler Sums

It is well known that sometimes Euler sums (i.e., alternating multiple zeta values) can be expressed as \Q-linear combinations of multiple zeta values (MZVs). In her thesis Glanois presented a criterion for a motivic Euler sums (MES) to be unramified, namely, expressible as \Q-linear combinations of motivic MZVs. By applying this criterion we present a few families of such unramified MES in two groups. In one such group we can further prove the concrete identities relating the MES to the motivic MZVs, determined up to rational multiple of a motivic Riemann zeta value by a result of Brown.Comment: 24 page

Variants of Multiple Zeta Values with Even and Odd Summation Indices

In this paper, we define and study a variant of multiple zeta values of level 2 (which is called multiple mixed values or multiple $M$-values, MMVs for short), which forms a subspace of the space of alternating multiple zeta values. This variant includes both Hoffman's multiple $t$-values and Kaneko-Tsumura's multiple $T$-values as special cases. We set up the algebra framework for the double shuffle relations (DBSFs) of the MMVs, and exhibits nice properties such as duality, integral shuffle relation, series stuffle relation, etc., similar to ordinary multiple zeta values. Moreover, we study several $T$-variants of Kaneko-Yamamoto type multiple zeta values by establishing some explicit relations between these $T$-variants and Kaneko-Tsumura $\psi$-values. Furthermore, we prove that all Kaneko-Tsumura $\psi$-values can be expressed in terms of Kaneko-Tsumura multiple $T$-values by using multiple associated integrals, and find some duality formulas for Kaneko-Tsumura $\psi$-values. We also discuss the explicit evaluations for a kind of MMVs of depth two and three by using the method of contour integral and residue theorem. Finally, we investigate the dimensions of a few interesting subspaces of MMVs for small weights.Comment: 35 page, 1 grap

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

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.Comment: 31 pages, section 1 revised to add connections to the Schur multiple zeta value

A Note on Sun's Conjectures on Ap\'ery-like Sums Involving Lucas Sequences and Harmonic Numbers

In this paper, we will prove Zhi-Wei Sun's four conjectural identities on Ap\'{e}ry-like sums involving Lucas sequences and harmonic numbers by using a few results of Davydychev--Kalmykov.Comment: 4 page

Alternating Multiple Mixed Values

In this paper, we define and study a variant of multiple zeta values (MZVs) of level four, called alternating multiple mixed values or alternating multiple $M$-values (AMMVs), which forms a subspace of the space of colored MZVs of level four as ${\mathbb Q}[i]$-vector spaces. This variant includes the alternating version of Hoffman's multiple $t$-values, Kaneko-Tsumura's multiple $T$-values, and the multiple $S$-values studied by the authors previously as special cases. We exhibit nice properties similar to the ordinary MZVs such as the duality, integral shuffle and series stuffle relations. After setting up the algebraic framework we derive the regularized double shuffle relations of the AMMVs. We also investigate several alternating multiple $T$- and $S$-values by establishing some explicit relations of integrals involving arctangent function. In the end, we discuss the explicit evaluations of a kind of AMMVs at depth three and compute the dimensions of a few interesting subspaces of AMMVs for weight less than 7.Comment: 40 pages, 3 figure

Berndt-Type Integrals of Order Three and Series Associated with Jacobi Elliptic Functions

In this paper, we first establish explicit evaluations of six classes of hyperbolic sums by special values of the Gamma function by using the tools of the Fourier series expansions and the Maclaurin series expansions of a few Jacobi elliptic functions developed in our previous paper. Then, using the method of contour integrations involving hyperbolic and trigonometric functions, we establish explicit evaluations of two families of Berndt-type integrals of order three by special values of the Gamma function. Furthermore, we present some interesting consequences and illustrative examples.Comment: 19 pages. arXiv admin note: text overlap with arXiv:2301.0821