540 research outputs found
Ramified and Unramified Motivic Multiple -, - and -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 -values, Kaneko and Tsumura's multiple -values, and the multiple
-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 -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
-values. Further, we are able to generalize a result of Charlton to more
families of unramified multiple -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 -values, MMVs for
short), which forms a subspace of the space of alternating multiple zeta
values. This variant includes both Hoffman's multiple -values and
Kaneko-Tsumura's multiple -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 -variants of Kaneko-Yamamoto type multiple zeta values by
establishing some explicit relations between these -variants and
Kaneko-Tsumura -values. Furthermore, we prove that all Kaneko-Tsumura
-values can be expressed in terms of Kaneko-Tsumura multiple -values
by using multiple associated integrals, and find some duality formulas for
Kaneko-Tsumura -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 where is a single-variable multiple polylogarithm
function or -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 -values, multiple
-values, multiple -values etc.), and multiple harmonic (star) sums and
their related variants (multiple -harmonic sums, multiple -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
-values (AMMVs), which forms a subspace of the space of colored MZVs of
level four as -vector spaces. This variant includes the
alternating version of Hoffman's multiple -values, Kaneko-Tsumura's multiple
-values, and the multiple -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 - and -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
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