Apéry-type series via colored multiple zeta values and Fourier-Legendre series expansions

In this paper, by applying the general Fourier-Legendre series expansion, we establish four general series transformations, and obtain a range of relations between the parametric Apéry-type series and the double sums of products of multiple harmonic sums (MHSs) or multiple t-harmonic sums (MtSs) from the Fourier-Legendre series expansions of the complete elliptic integrals of the first and second kind as well as two special expansions provided recently in the literature. By establishing the linearization theorem for the double sums of products above, and using the methods of partial fraction decomposition and transformation of summations, we show that these parametric Apéry-type series are expressible in terms of some elementary series involving MHSs and MtSs, and finally reducible, with an extra factor $1/\pi$, to linear combinations of alternating multiple zeta values and colored multiple zeta values of level four. By specifying the parameters, we determine the evaluations of many special Apéry-type series.