Fibonacci-Motzkin paths and J-type continued fractions Recall that a Motzkin path is a piece-wise linear planar path
connecting points in the integer lattice quadrant
$\Bbb{Z}_{\geq 0} \times \Bbb{Z}_{\geq 0}$ beginning at the origin $(0,0)$ and
ending at $(n,0)$ for some $n \in \Bbb{Z}_{>0}$ whose steps
are either
\begin{equation}
\begin{array}{ll}
\nearrow & = (1,1) \\
\searrow & = (1,-1) \\
\rightarrow & = (1,0)
\end{array}
\end{equation}
Alternatively we may view a Motzkin path as
an $n$-tuple $\underline{\pi} = \big(\pi_1, \dots, \pi_n \big)$
where $\pi_i \in \{ \nearrow \, , \searrow \, , \rightarrow \}$
for each $1 \leq i \leq n$
and where $\pi_1 + \cdots + \pi_n = (n,0)$. The set
of Motzkin paths $\underline{\pi}$ terminating at $(n,0)$ will be denoted $\mathcal{M}_n$. Among the Motzkin paths $\mathcal{M}_n$
is the subset of what I'll call Fibonacci-Motzkin paths: These are paths that weakly increase until a threshold is reached, after
which they strictly decrease. More specifically
$\underline{\pi} = (\pi_1, \dots, \pi_n) \in \mathcal{M}_n$
is a Fibonacci-Motzkin path if there exists a threshold
$ 0 \leq k \leq \lfloor {1 \over 2} n \rfloor  $
such that $\pi_i \in \{ \nearrow, \, \rightarrow \}$ for
all $1 \leq i \leq n - k$ and $\pi_i = \searrow$ for all $n-k < i \leq n$. The threshold $k$ equals the number of $\nearrow$ steps
taken in the initial ascent of the path.
A moment's reflection should convince the reader that
the cardinality of the set $\mathcal{F}_n$ of all
Fibonacci-Motzkin paths is indeed the $n$-th
Fibonacci number, thus justifying the choice of terminology.
Received wisdom tells us to introduce
two infinite families of generic parameters $\beta_1, \beta_2, \beta_3, \dots$ and $\gamma_0, \gamma_1, \gamma_2, \dots$ and then
assign a weight
\begin{equation}
\mathrm{w}(\underline{\pi}) \, := \ \beta_1 \cdots
\beta_k \cdot \gamma_0^{m_0} \cdots \gamma_k^{m_k} 
\end{equation}
to a Motzkin path $\underline{\pi} \in \mathcal{M}_n$
where
\begin{equation}
\begin{array}{ll}
k  
&\text{$=$ number of $\nearrow$ steps taken by $\underline{\pi}$} \\
m_\ell 
&\text{$=$ number of $\rightarrow$ steps taken by $\underline{\pi}$ at height $\ell$} 
\end{array}
\end{equation}
The corresponding generating function of all Motzkin paths, given by
\begin{equation}
M(z) := \ 1 \ + \ \sum_{n \, \geq \, 1} z^n \sum_{\underline{\pi} \, \in \, \mathcal{M}_n} \, \mathrm{w}(\underline{\pi}) 
\end{equation}
is then seen to coincide with the formal expansion of the
$J$-type continued fraction
\begin{equation}
\ {1 \over {1 - z \gamma_0 \ - \ {\displaystyle z^2 \beta_1 \over {\displaystyle 1 - z \gamma_1 \ - \ {z^2 \beta_2 \over {\displaystyle 1 - z \gamma_2 \ - \ {z^2 \beta_3 \over {\ddots}}}}}}}}
\end{equation}
Let us introduce a Fibonacci analogue of the generating
function $M(z)$ namely
\begin{equation}
F(z) := \ 1 \ + \ \sum_{n \, \geq \, 1} z^n \sum_{\underline{\pi} \, \in \, \mathcal{F}_n} \, \mathrm{w}(\underline{\pi}) 
\end{equation}
Question: Is there some kind of continued fraction whose
expansion is $F(z)$?
thanks, ines.
 A: Grouping the terms of $F(z)$ by the height reached, we get
$$F(z) = \frac{1}{(1 - z\gamma_0)} + \frac{z^2 \beta_1}{(1 - z\gamma_0) (1 - z\gamma_1)} + \frac{z^4 \beta_1 \beta_2}{(1 - z\gamma_0) (1 - z\gamma_1) (1 - z\gamma_2)} + \cdots \\
$$
This has the form of Euler's continued fraction $$a_0 + a_0a_1 + a_0a_1a_2 + \cdots = \cfrac{a_0}{1 - \cfrac{a_1}{1 + a_1 - \cfrac{a_2}{1 + a_2 - \ddots}}}$$
with $$a_0 = \frac{1}{1 - z\gamma_0} \\
a_1 = \frac{z^2 \beta_1}{1 - z \gamma_1} \\
a_2 = \frac{z^2 \beta_2}{1 - z \gamma_2} \\
\vdots
$$
It is perhaps more natural to drop the denominators to the next level: i.e. instead of $$\cfrac{\frac{1}{1 - z\gamma_0}}{1 - \cfrac{\frac{z^2 \beta_1}{(1 - z \gamma_1)}}{1 + \frac{z^2 \beta_1}{(1 - z \gamma_1)} - \cfrac{\frac{z^2 \beta_2}{(1 - z \gamma_2)}}{1 + \frac{z^2 \beta_2}{(1 - z \gamma_2)} - \ddots}}}$$ we could write $$\cfrac{1}{(1 - z\gamma_0) - \cfrac{(1 - z\gamma_0) z^2 \beta_1}{(1 - z \gamma_1) + z^2 \beta_1 - \cfrac{(1 - z \gamma_1) z^2 \beta_2}{(1 - z \gamma_2) + z^2 \beta_2 - \ddots}}}$$
