Apéry-type series and colored multiple zeta values

Abstract

In this paper, we study new classes of Apéry-type series involving the central binomial coefficients and the multiple t-harmonic sums by combining the methods of iterated integrals 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 Apéry-type series involving harmonic sums to those with products of multiple harmonic sums and multiple t-harmonic sums. We show that these series can be expressed as either the real or the imaginary part of a Q-linear combination of colored multiple zeta values of level 4. Hopefully, these relations will shed some new lights on their properties which may lead to novel approaches to irrationality questions on the Riemmann zeta values, or more generally, the multiple zeta values.