Ap\'ery-Type Series and Colored Multiple Zeta Values
Ce Xu, Jianqiang Zhao

TL;DR
This paper introduces new Apéry-type series involving multiple t-harmonic sums, connecting them to colored multiple zeta values of level 4 through advanced integral and series expansion methods.
Contribution
It develops a novel approach combining iterated integrals and Fourier--Legendre expansions to relate Apéry-type series with colored multiple zeta values of level 4.
Findings
Series can be expressed as real or imaginary parts of Q-linear combinations of colored multiple zeta values of level 4.
Generalization of old Apéry-type series to include products of harmonic sums and multiple t-harmonic sums.
Establishment of a connection between Apéry-type series and multiple zeta values using new methods.
Abstract
In this paper, we introduce and study new classes of Ap\'ery-type series involving the multiple -harmonic sums by combining the methods of iterated integral and Fourier--Legendre series expansions, where the multiple -harmonic sums are a variation of multiple harmonic sums in which all the summation indices are restricted to odd numbers only. Our approach also enables us to generalize some old classes of Ap\'ery-type series involving harmonic sums to those with products of harmonic sums and multiple -harmonic sums. In all of these series, the central binomial coefficients appear as or where . We show that every such series can be expressed as either the real or the imaginary part of a -linear combination of colored multiple zeta values of level 4.
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Taxonomy
TopicsAdvanced Mathematical Identities · Analytic Number Theory Research · Advanced Combinatorial Mathematics
