**Background** (may be skipped by those interested only in the basic question and not important associations):
“[An essay on continued fractions][1]” by Euler (translated by Myra and Bostwick Wyman) contains on page 319 Euler's continued fraction
$$
\begin{split} q & = \frac{a}{p} + \frac{1}{3\dfrac{a}{p}+\dfrac{1 }{5\dfrac{a}{p}+\dfrac{1}{7\dfrac{a}{p}+\dfrac{ 1}{\ddots}} } \; \;}\\
&= CF\left[\;\frac{a}{p}, \; 3\frac{a}{p}, \; 5 \frac{a}{p}, ..., \; (2n+1)\frac{a}{p}, \; ... \;\right].
\end{split}
$$
The CF for the reciprocal is
$$ \bar{q} = \frac{1}{q} = CF\left[0,\;\frac{a}{p}, \; 3\frac{a}{p}, \; 5 \frac{a}{p}, ..., \; (2n+1)\frac{a}{p}, \; ... \;\right].$$
Euler shows that the value of $q$ is defined by the Riccati equation
$$a \; dq + q^2 dp = dp.$$
$$ \frac{dq}{dp} = q'(p) = \frac{1-q^2}{a} .$$
Then from the formulas in my contribution (Sept 18, 2014) to OEIS [008292][2] on the Eulerian numbers with $\hat{p} = \frac{p}{a}$,
**1)**
$$ \frac{1}{q} = \bar{q} = \frac{e^{\hat{p}}-e^{-\hat{p}}}{e^{\hat{p}}+e^{-\hat{p}}} = \tanh(\hat{p}) =A(\hat{p}, \;1, \; -1),$$
where
$$A(x,a,b)= \frac{e^{ax}-e^{bx}}{a\;e^{bx}-b\;e^{ax}} = x + (a+b) \;\frac{x^2}{2!} + (a^2+4\;ab+b^2)\; \frac{x^3}{3!} + (a^3+11\;a^2b+11\;ab^2+b^3) \; \frac{x^4}{4!} + ...$$
is an e.g.f. for the bivariate Eulerian polynomials $E_n(a,b)$, whose coefficients are those of the h-vectors for the permutohedra,
**2)**
$$ \frac{p}{a} = \hat{p} = \frac{1}{2} \; \ln\left[ \;\frac{1+\bar{q}}{1-\bar{q}} \;\right] = \tanh^{(-1)}(\bar{q}) = B(\bar{q}, \;1, \; -1),$$
where
$$ B(x,a,b) = \frac{1}{a-b} \; \ln\left[ \; \frac{1+ax}{1+bx} \;\right] = x - (a+b) \; \frac{x^2}{2} + (a^2+ab+b^2)\; \frac{x^3}{3} - (a^3+a^2b+ab^2+b^3) \; \frac{x^4}{4} + ... $$
$$= \ln(1+u.\;x),$$
with $(u.)^n = u_n = h_{n-1}(a,b)$ a complete homogeneous polynomial in two indeterminates with $h_n(1,x)$ the h-vector of the ${(n-1)}$-dimensional hypertetrahedron, is a log generating function for the complete homogeneous polynomials,
**3)**
$$ \frac{d\bar{q}}{d\hat{p}} = \bar{q}' = (1+\bar{q})\;(1-\bar{q}),$$
an instance of the Riccati equation
$$ D_x \; A(x,a,b) = A'(x,a,b) = (1+a \;A)\;(1+b\;A),$$
which can be written in terms of a Weierstrass elliptic function (see Buchstaber & Bunkova in the OEIS entry)
**4)**
$$ \frac{1}{q} = \bar{q}(\hat{p}) = e^{\hat{p}\; (1-u^2) \; D_u} \; u \; |_{u=0},$$
more generally the bivariate Eulerian row polynomials $E_n(a,b)$ of $A(x,a,b)$ with $E_0(a,b) =0$ are generated by
$$ E_n(a,b) = [\;(1+ax) \; (1+bx) \; D_x\;]^{n} \; x \; |_{x=0}$$
(see OEIS [A145271][3] for a generator of compositional inverses via the refined Eulerian numbers)
So, with $x = a/p$ the continued fraction of Euler evaluates analytically as
$$q(x) = \coth\left(\frac{1}{x}\right) = \coth\left(\frac{p}{a}\right)$$
with a discontinuity--a jump from $-1$ to $1$ as the argument passes through the origin from negative to positive values of $x$.
The reciprocal, of course, is
$$\bar{q} = \frac{1}{q} = \tanh\left(\frac{1}{x}\right) = \tanh\left(\frac{p}{a}\right)$$
with the same discontinuity at the origin $x=0$.
The more natural presentation is
$$\bar{q} = \tanh\left(\frac{p}{a}\right)$$
with no discontinuity for finite real argument $\frac{p}{a}$.
Note:
$$FGL(x,y) = A(B(x,a,b) + B(y,a,b),a,b) = \frac{x+y+(a+b)xy}{1-ab\; xy}$$
is called the hyperbolic formal group law and related to a generalized cohomology theory proposed by Lenart and Zainoulline.
For $a=-b=1$
$$FGL_{\tanh}(x,y) = \frac{x+y}{1+\; xy}$$
This is the addition, or composition, law for velocities in special relativity for $c=1$ and the formula for the hyperbolic tangent of sums
$$ \tanh(z+w) = \frac{\tanh(z)+\tanh(w)}{1+\tanh(z) \tanh(w)}.$$
See my post "[The Elliptic Lie Triad: KdV and Riccati Equations, Infinigens, and Elliptic Genera][4]" for relationships to a soliton solution to the KdV equation and an associated Riccati equation or my contribution to the MO-Q "[Is there an underlying explanation for the magical powers of the Schwarzian?][5]" for a briefer note on some aspects of the relationships.
**Question**:
What are continued fraction reps for
$$A(x,a,b)= \frac{e^{ax}-e^{bx}}{a\;e^{bx}-b\;e^{ax}} = x + (a+b) \;\frac{x^2}{2!} + (a^2+4\;ab+b^2)\; \frac{x^3}{3!} + (a^3+11\;a^2b+11\;ab^2+b^3) \; \frac{x^4}{4!} + ...$$
and what references for any specific rep are available (via the usual free sources)?
I suspect some version of Equation 4 in "[Introduction to Chapter 3 on continued fractions][6] [version 5, 29 January 2013]" by Xavier Viennot interpreted in terms of Dyck lattice paths should apply since the associahedra partition polynomials of OEIS [A133437][7] for compositional inversion can be applied to $B(x,a,b)$ to obtain $A(x,a,b)$ and these associahedra face polynomials are a refinement of those of [A126216][8], which are related to marked Dyck paths (and Schroeder--see Drake therein).
Another potential lead is [A134264][9] / [A125181][10] for compositional inversion via noncrossing partitions / Dyck paths of even length. See "[A note on 2-distant noncrossing partitions and weighted Motzkin paths][11]" by Ira Gessel and Jang Soo Kim, related to CFs.
EDIT 7/3/21:
I've scanned over dozens of references on orthogonal polynomials and continued fractions over the last couple of weeks, but didn't come across until just now "[Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes–Rogers and Thron–Rogers polynomials, with coefficientwise Hankel-total positivity][12]" by Mathias Petreolle, Alan Sokal, and Bao-Xuan Zhu. The footnote on p.77 states:
*The identity (12.6) — that is, the S-fraction for the Eulerian polynomials — was found by Stieltjes [160, section 79]. Stieltjes does not specifically mention the Eulerian polynomials, but he does state that the continued fraction is the formal Laplace transform of $(1 − y)/(e^{t(y−1)} − y), which is well known to be the exponential generating function of the Eulerian polynomials. Stieltjes also refrains from showing the proof: “Pour abreger, je supprime toujours les artifices qu’il faut employer pour obtenir la transformation de l’int´egrale definie en fraction continue” (!). But a proof is sketched, albeit also without much explanation, in the book of Wall [165, pp. 207–208]. The J-fraction corresponding to the contraction of this S-fraction was proven, by combinatorial methods,
by Flajolet [52, Theorem 3B(ii) with a slight typographical error]. Dumont [41, Propositions 2 and 7] gave a direct combinatorial proof of the S-fraction, based on an interpretation of the Eulerian polynomials in terms of “bipartite involutions of [2n]” and a bijection of these onto Dyck paths.*
(The paper also contains, on p. 83, the partition polynomials of [A190015][13], which they call the Euler symmetric polynomials, claiming to have introduced them originally in their paper. As noted in the OEIS entry, they are a scaled version of A145271, which I call the refined Eulerian partition polynomials mentioned above.)
Dumont's paper "[Pics de cycle et derivees partielles][14]" gives, on p. 38, the continued fraction for an o.g.f. of the bivariate Eulerian polynomials as
$$x + \sum_{n \geq 1} A_n(x,y)\; u^n $$
$$
\begin{split} & = \frac{x}{1-\dfrac{yu }{1-\dfrac{xu}{1-\dfrac{ 2yu}{\ddots}} } \; \;}\\
&= SFC[\; x, \; \frac{yu}{xu}, \; \frac{2yu}{2xu}, \; \frac{3yu}{3xu},..]
\end{split}
$$
so the question becomes what is the continued fraction for an e.g.f. given the continued fraction of its Laplace transform, or shifted o.g.f.?
[1]: https://www.researchgate.net/publication/227115346_An_essay_on_continued_fractions_-_Leonhard_Euler
[2]: https://oeis.org/A008292
[3]: https://oeis.org/A145271
[4]: https://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/
[5]: https://mathoverflow.net/questions/38105/is-there-an-underlying-explanation-for-the-magical-powers-of-the-schwarzian-deri/221682#221682
[6]: https://www.stat.purdue.edu/~mdw/ChapterIntroductions/ContinuedFractionsUpdateViennot.pdf
[7]: https://oeis.org/A133437
[8]: https://oeis.org/A126216
[9]: https://oeis.org/A134264
[10]: https://oeis.org/A125181
[11]: https://arxiv.org/abs/1003.5301
[12]: http://arxiv.org/abs/1807.03271
[13]: https://oeis.org/A190015
[14]: http://emis.dsd.sztaki.hu/journals/SLC/opapers/s13dumont.pdf