Uniform treatments of Bernoulli numbers, Stirling numbers, and their generating functions

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 Bn,k, the author establishes thirteen Maclaurin series expansions of the functions {align*} &^x+1/2, && ^x-1/x, && ln x, \\ &ln x/x, && [ln(1+x)/x]^r, && (e^x-1/x)^r {align*} for r=±1/2 and r in terms of the Dirichlet eta function η(1-2k), the Riemann zeta function ζ(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 B₂ₙ. finds out three closed-form formulas for the Bernoulli numbers B₂ₙ 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 $S(n,k)$.