Apéry-Type Series and Colored Multiple Zeta Values

Abstract: In this paper, we introduce and study new classes of Ap\'ery-type series involving the multiple $t$-harmonic sums by combining the methods of iterated integral and Fourier--Legendre series expansions, where the multiple $t$-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 $t$-harmonic sums. In all of these series, the central binomial coefficients appear as $a_n^{\pm 1}$ or $a_n^{\pm 2}$ where $a_n=\binom{2n}{n}/4^n$. We show that every such series can be expressed as either the real or the imaginary part of a $\mathbb Q$-linear combination of colored multiple zeta values of level 4.