Integral Representations for Multiple Apéry-Like Series

We derive integral representations for six families of multiple Apéry-like series using repeated integration by parts and Fourier expansions. The resulting formulas are expressed in terms of polylogarithms, Legendre chi functions, and inverse tangent integrals. As applications, we recover several known evaluations as special cases of our results, expressed in terms of Dirichlet eta, beta, and lambda functions. In addition, we obtain a new identity expressing a family of such series as linear combinations of products of Dirichlet eta values.