Uniform treatments of Bernoulli numbers, Stirling numbers, and their generating functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909591064805376 |
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| author | Qi, Feng |
| author_facet | Qi, Feng |
| contents | 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 $S(n,k)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_16965 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform treatments of Bernoulli numbers, Stirling numbers, and their generating functions Qi, Feng Combinatorics Classical Analysis and ODEs Primary 05A15, Secondary 03D20, 11B73, 11B83, 26A09, 33B10, 41A58 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 $S(n,k)$. |
| title | Uniform treatments of Bernoulli numbers, Stirling numbers, and their generating functions |
| topic | Combinatorics Classical Analysis and ODEs Primary 05A15, Secondary 03D20, 11B73, 11B83, 26A09, 33B10, 41A58 |
| url | https://arxiv.org/abs/2504.16965 |