This 2021 post discusses the continued fraction, $$F_4=1+\cfrac{1/1}{1+\cfrac{1/3}{1+\cfrac{1/5}{1+\cfrac{1/7}{1+\ddots}}}}= \frac{2}{ 1+\displaystyle\frac{I_1(1/4)}{I_0(1/4)}}=1.7793\dots$$
where $I_n(x)$ is the "modified Bessel function of the first kind". Using a transformation, we find this is equivalent to the equally nice,
$$F_4=1+\cfrac{1}{1+\cfrac{1}{3+\cfrac{3}{5+\cfrac{5}{7+\ddots}}}}= \frac{2}{ 1+\displaystyle\frac{I_1(1/4)}{I_0(1/4)}}=1.7793\dots$$
Defining the function,
$$F_k = \frac{2}{ 1+\displaystyle\frac{I_1(1/k)}{I_0(1/k)}}\quad$$
we get the simple continued fraction expansions via Wolfram,
\begin{align} F_2&=(1, 1, \color{blue}1, 1, 1, \color{blue}3, 1, 1, \color{blue}{5}, 1, 1, \color{blue}{7}, 1, 1, \color{blue}{9}, 1, 1,\dots)\\ F_3&=(1, 1, \color{blue}2, 1, 1, \color{blue}5, 1, 1, \color{blue}{8}, 1, 1, \color{blue}{11}, 1, 1, \color{blue}{14}, 1, 1,\dots)\\ F_4&=(1, 1, \color{blue}3, 1, 1, \color{blue}7, 1, 1, \color{blue}{11}, 1, 1, \color{blue}{15}, 1, 1, \color{blue}{19}, 1, 1,\dots)\end{align}
where we use all commas for simplicity. But I've seen similar regular behavior before, in $e$,
\begin{align} 1+1/e^{1/2}&=(1, 1, \color{blue}1, 1, 1, \color{blue}5, 1, 1, \color{blue}{9}, 1, 1, \color{blue}{13}, 1, 1, \color{blue}{17}, 1, 1,\dots)\\ 1+1/e^{1/3}&=(1, 1, \color{blue}2, 1, 1, \color{blue}8, 1, 1, \color{blue}{14}, 1, 1, \color{blue}{20}, 1, 1, \color{blue}{26}, 1, 1,\dots)\\ 1+1/e^{1/4}&=(1, 1, \color{blue}3, 1, 1, \color{blue}{11}, 1, 1, \color{blue}{19}, 1, 1, \color{blue}{27}, 1, 1, \color{blue}{35}, 1, 1,\dots)\end{align}
Note the "natural" cfracs of the Bessel ratios,
$$P_m =\cfrac{1}{1m +\cfrac{1}{2m + \cfrac{1}{3m + \ddots}}} = \frac{I_1(2/m)}{I_0(2/m)} = \frac{2}{F_{m/2}-1} = \;??\quad $$
$$Q_m =\cfrac{1}{1m +\cfrac{1}{3m + \cfrac{1}{5m + \ddots}}} = \frac{I_{1/2}(1/m)}{I_{-1/2}(1/m)} =\frac{e^{2/m}-1}{e^{2/m}+1} $$
thus we can express $e^{1/n}$ in terms of the second Bessel ratio $\dfrac{I_{1/2}(u)}{I_{-1/2}(u)}$.
Q: So can we also express $e^{1/n}$ in terms of the first Bessel ratio $\dfrac{I_{1}(v)}{I_{0}(v)}$?
