Apéry-type series via Bell polynomials

In this paper, we develop a systematic theory based on Bell polynomials for evaluating parametric Apéry-type series. More specifically, starting from known series expansions involving central binomial coefficients, we establish connections between parametric Apéry-type series and beta-type integrals or generalized log-sine integrals. By applying several lemmas involving Bell polynomials to these special integrals and using properties of polygamma function and other special functions, we calculate these integrals and, consequently, the corresponding Apéry-type series. By specifying the parameters, we obtain evaluations for many special cases of such Apéry-type series.