In this paper, by virtue of a determinantal formula for derivatives of the ratio between two differentiable functions, in view of the Fa\`a di Bruno formula, and with the help of several identities and closed-form formulas for the partial Bell polynomials B₍, ₊, the author establishes thirteen Maclaurin series expansions of the functions align* &eˣ+12, && eˣ-1x, && x, \\ & xx, && (1+x) xʳ, && (eˣ-1x) ʳ align* for r=12 and r in terms of the Dirichlet eta function (1-2k), the Riemann zeta function (1-2k), and the Stirling numbers of the first and second kinds s (n, k) and S (n, k). presents four determinantal expressions and three recursive relations for the Bernoulli numbers B₂₍. finds out three closed-form formulas for the Bernoulli numbers B₂₍ and the generalized Bernoulli numbers Bₙ^ (r) in terms of the Stirling numbers of the second kind S (n, k), and deduce two combinatorial identities for the Stirling numbers of the second kind S (n, k). acquires two combinatorial identities, which can be regarded as diagonal recursive relations, involving the Stirling numbers of the first and second kinds s (n, k) and S (n, k). recovers an integral representation and a closed-form formula, and establish an alternative explicit and closed-form formula, for the Bernoulli numbers of the second kind bₙ in terms of the Stirling numbers of the first kind s (n, k). obtains three identities connecting the Stirling numbers of the first and second kinds s (n, k) and S (n, k).
Feng Qi (Wed,) studied this question.