**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)? 


  [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