It is known that
\begin{equation\*}
\tan x=\sum\_{k=1}^{\infty}\frac{2^{2k}\bigl(2^{2k}-1\bigr)}{(2k)!}|B\_{2k}|x^{2k-1}, \quad |x|<\frac{\pi}{2}
\end{equation\*}
and
\begin{equation\*}
\ln\tan x=\ln x+\sum\_{k=1}^{\infty}\frac{2^{2k}\bigl(2^{2k-1}-1\bigr)}{k(2k)!}|B\_{2k}|x^{2k}, \quad 0<x<\frac{\pi}{2},
\end{equation\*}
where $B\_{2k}$ denotes the classical Bernoulli numbers. From the second series expansion, we acquire
\begin{equation\*}
\ln\frac{\tan x}{x}=\sum\_{k=1}^{\infty}\frac{2^{2k}\bigl(2^{2k-1}-1\bigr)}{k(2k)!}|B\_{2k}|x^{2k}, \quad |x|<\frac{\pi}{2}.
\end{equation\*}
My question is:


What and where is the series expansion of the function $\ln\bigl(\frac{\tan x}{x}-1\bigr)$ or $\ln(\tan x-x)$ around $x=0$?


References


1. I. S. Gradshteyn and I. M. Ryzhik, *Table of Integrals, Series, and Products*, Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Eighth edition, Revised from the seventh edition, Elsevier/Academic Press, Amsterdam, 2015; available online at <https://doi.org/10.1016/B978-0-12-384933-5.00013-8>.



What or where is the series expansion of the function $\ln\bigl(\frac{\tan x}{x}-1\bigr)$ or $\ln(\tan x-x)$ around $x=0$?
Let
\begin{equation\*}
f(x)=\begin{cases}
\ln\dfrac{\tan x-x}{x^3}, & 0<|x|<\dfrac{\pi}{2};\\
-\ln3, & x=0.
\end{cases}
\end{equation\*}
Then the even function $f(x)$ has the Maclaurin power series expansion
\begin{equation}\label{ln-tan-x-cubic-ser-expans}
\begin{aligned}
f(x)&=-\ln3-\sum\_{k=1}^{\infty}\frac{3^{2k}D\_{2k}}{(2k)!}x^{2k}\\
&=-\ln3+\frac{2 x^2}{5}+\frac{43 x^4}{525}+\frac{524 x^6}{23625}+\frac{40897 x^8}{6063750} +\frac{19393844 x^{10}}{8868234375}+\dotsm
\end{aligned}
\end{equation}
for $|x|<\frac{\pi}{2}$, where
\begin{align\*}
D\_{2k}&=\begin{vmatrix}
0 & Q\_0 & 0 & \dotsm&0& 0& 0\\
Q\_1 & 0 & Q\_0 & \dotsm& 0&0& 0\\
0 & Q\_1 & 0 & \dotsm& 0& 0&0\\
\dotsm & \dotsm & \dotsm & \ddots&\dotsm& \dotsm& \dotsm\\
Q\_{k-1} & 0 & \binom{2k-3}{1}Q\_{k-2} & \dotsm& 0 & Q\_0 & 0\\
0 & Q\_{k-1} & 0 & \dotsm& \binom{2k-2}{2k-4}Q\_1 & 0 & Q\_0\\
Q\_{k} & 0 & \binom{2k-1}{1}Q\_{k-1} & \dotsm& 0 & \binom{2k-1}{2k-3}Q\_1 & 0
\end{vmatrix}\\
&=\begin{vmatrix}
e\_{i,j}
\end{vmatrix}\_{2k\times2k}, \quad k\ge1,\\
e\_{i,j}&=\begin{cases}
0, & (i,j)=(2\ell-1,1),\quad 1\le\ell\le k;\\
Q\_\ell, & (i,j)=(2\ell,1),\quad 1\le\ell\le k;\\
0, & 1\le i\le j-2\le2k-2;\\
0, & i-j=2\ell, \quad 0\le\ell\le k-1;\\
\binom{i-1}{j-2}Q\_{\ell}, & i-j=2\ell-1, \quad 0\le\ell\le k-1,\quad j\ge2,
\end{cases}\\
Q\_m&=\frac{2^{2m+2}\bigl(2^{2m+4}-1\bigr)}{(m+1)(m+2)(2m+1)(2m+3)}|B\_{2m+4}|, \quad m\ge0,
\end{align\*}
and $B\_{2m+4}$ denotes the Bernoulli numbers generated by
\begin{equation\*}
\frac{z}{\operatorname{e}^z-1}=\sum\_{k=0}^\infty B\_k\frac{z^k}{k!}=1-\frac{z}2+\sum\_{k=1}^\infty B\_{2k}\frac{z^{2k}}{(2k)!}, \quad |z|<2\pi.
\end{equation\*}
As corollaries of the above Maclaurin power series expansion, we deduce the series expansions
\begin{equation\*}
\ln(\tan x-x)=-\ln3+3\ln x-\sum\_{k=1}^{\infty}\frac{3^{2k}D\_{2k}}{(2k)!}x^{2k}, \quad 0<x<\frac{\pi}{2}
\end{equation\*}
and
\begin{equation\*}
\ln\biggl(\frac{\tan x}{x}-1\biggr)
=-\ln3+2\ln|x|-\sum\_{k=1}^{\infty}\frac{3^{2k}D\_{2k}}{(2k)!}x^{2k}, \quad 0<|x|<\frac{\pi}{2}.
\end{equation\*}
Since the Bernoulli numbers can be expressed by closed-form formulas, so the sequence $D\_{2k}$ is surely a closed-form formula.


About closed-form formulas for the Bernoulli numbers and about some conclusions on the tangent function, I would like to recommend the following papers.


References


1. Xue-Yan Chen, Lan Wu, Dongkyu Lim, and Feng Qi, *Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind*, Demonstratio Mathematica **55** (2022), no. 1, 822--830; available online at <https://doi.org/10.1515/dema-2022-0166>.
2. Feng Qi and Jacques Gelinas, *Revisiting Bouvier's paper on tangent numbers*, Advances and Applications in Mathematical Sciences **16** (2017), no. 8, 275--281.
3. Feng Qi and Bai-Ni Guo, *An explicit formula for derivative polynomials of the tangent function*, Acta Universitatis Sapientiae Mathematica **9** (2017), no. 2, 348--359; available online at <https://doi.org/10.1515/ausm-2017-0026>.
4. Feng Qi, *Derivatives of tangent function and tangent numbers*, Applied Mathematics and Computation **268** (2015), 844--858; available online at <https://doi.org/10.1016/j.amc.2015.06.123>.
5. Jiao-Lian Zhao, Qiu-Ming Luo, Bai-Ni Guo, and Feng Qi, *Remarks on inequalities for the tangent function*, Hacettepe Journal of Mathematics and Statistics **41** (2012), no. 4, 499--506.
6. Chao-Ping Chen and Feng Qi, *A double inequality for remainder of power series of tangent function*, Tamkang Journal of Mathematics **34** (2003), no. 4, 351--355; available online at <https://doi.org/10.5556/j.tkjm.34.2003.236>.
7. <https://math.stackexchange.com/a/4248328>
8. <https://math.stackexchange.com/a/4248341>
9. <https://math.stackexchange.com/a/4248488>
10. <https://math.stackexchange.com/a/4254493>
11. <https://math.stackexchange.com/a/4254500>
12. <https://math.stackexchange.com/a/4256893>
13. <https://math.stackexchange.com/a/4256913>
14. <https://math.stackexchange.com/a/4256915>
15. <https://math.stackexchange.com/a/4656534>


