Uniform treatments of Bernoulli numbers, Stirling numbers, and their generating functions
Feng Qi
TL;DR
The paper unifies Bernoulli and Stirling-number frameworks via a determinant-based derivative scheme anchored in the Faà di Bruno formula and Bell polynomials, delivering thirteen Maclaurin expansions of log- and power-type functions expressed through $\eta$, $\zeta$, and Stirling numbers. It provides multiple new determinantal expressions and recursive relations for $B_{2n}$, plus closed-form representations of $B_{2n}$ and $B_n^{(r)}$ in terms of $S(n,k)$, accompanied by several identities linking $s(n,k)$ and $S(n,k)$. It also develops generating functions for Stirling numbers of both kinds, yielding diagonal-type recurrences and unifications that connect Bernoulli-type numbers with Stirling structures through integral representations and Bell polynomials. Collectively, these results deepen the algebraic and combinatorial connections between Bernoulli numbers and Stirling numbers, offering new closed forms and identities with potential utility in analytic number theory and symbolic computation.
Abstract
In this paper, by virtue of a determinantal formula for derivatives of the ratio between two differentiable functions, in view of the Faà di Bruno formula, and with the help of several identities and closed-form formulas for the partial Bell polynomials $\operatorname{B}_{n,k}$, the author establishes thirteen Maclaurin series expansions of the functions \begin{align*} &\ln\frac{\operatorname{e}^x+1}{2}, && \ln\frac{\operatorname{e}^x-1}{x}, && \ln\cosh x, \\ &\ln\frac{\sinh x}{x}, && \biggl[\frac{\ln(1+x)}{x}\biggr]^r, && \biggl(\frac{\operatorname{e}^x-1}{x}\biggr)^r \end{align*} for $r=\pm\frac{1}{2}$ and $r\in\mathbb{R}$ 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 $B_{2n}$. finds out three closed-form formulas for the Bernoulli numbers $B_{2n}$ and the generalized Bernoulli numbers $B_n^{(r)}$ 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 $b_n$ 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 $S(n,k)$.