Is the ratio of consecutive Bernoulli polynomials uniformly bounded When investigating a certain kind of Stirling's approximation of the Gamma function error terms occur such as
\begin{equation}
E(s)=\frac{1}{s}\sum_{j=1}^\infty B_{j+1}(a)\frac{(-1)^{j+1}}{j(j+1)s^{j-1}},|s|\to\infty
\end{equation}
where $a,s\in\mathbb{C}$ and the $B_j$ are the Bernoulli polynomials. I'd like to survey the convergence properties of the sum consideraring the ratio
\begin{equation}
\left|\frac{B_{j+1}(a)}{B_j(a)}\frac{j-1}{j+1}\frac1s\right|.
\end{equation}
So the first question is whether $B_{j+1}(a)/B_j(a)$ is bounded for $j\to\infty$ and fixed $a$. This would yield $E(s)=O(1/|s|),|s|\to\infty$.
Now let $a\in K\subset\mathbb{C}$ with $K$ a compact set. Is $B_{j+1}(a)/B_j(a)$ bounded for $j\to\infty$ uniformly in $a\in K$?
Or maybe someone proposes another approach to get
\begin{equation}
E(s)=O(1/|s|),|s|\to\infty\text{ uniformly in $K$}
\end{equation}
$K=\{a\}$ resp. $K$ compact.
 A: In the paper [1] below, the following increasing property for the ratio of two non-zero neighbouring Bernoulli numbers $B_{2n}$ were obtained:

*

*The sequence $\frac{|B_{2(n+1)}|}{|B_{2n}|}$ is increasing in $n\in\mathbb{N}_0=\{0,1,2,\dotsc\}$ and tends to $+\infty$ as $n\to+\infty$. Consequently, the sequence $|B_{2n}|$is logarithmically convex in $n\in\mathbb{N}_0$.

*For fixed $\ell\in\mathbb{N}=\{1,2,3,\dotsc\}$, the sequence
\begin{equation}\label{Bernou-ratio-frac-seq}
\frac{\prod_{k=1}^{\ell}[2(n+1)+k]}{\prod_{k=1}^{\ell}(2n+k)}\frac{|B_{2(n+1)}|}{|B_{2n}|}
\end{equation}
is increasing in $n\in\mathbb{N}$ and tends to $+\infty$ as $n\to+\infty$. Consequently, for fixed $\ell\in\mathbb{N}$, the sequence
\begin{equation}\label{Bernou-frac-seq-log-conv}
\frac{(2n+\ell)!}{(2n)!}|B_{2n}|
\end{equation}
is logarithmically convex in $n\in\mathbb{N}$.

I think these increasing properties established in the paper [1] below are an answer to this question.
Reference

*

*Y. Shuang, B.-N. Guo, and F. Qi, Logarithmic convexity and increasing property of the Bernoulli numbers and their ratios, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115 (2021), no. 3, Paper No. 135, 12 pages; available online at https://doi.org/10.1007/s13398-021-01071-x.

