Let
\begin{equation} \begin{array}{ll} \displaystyle F(z) &\displaystyle = \ 1 \ + \ z \ + \ 2z^2 \ + \ 3z^3 \ + \ 5z^4 \ + \ \cdots \\ &\displaystyle = \ {1 \over {1 -z - z^2}} \end{array} \end{equation}
be the generating function of the Fibonacci numbers (along with its presentation as a rational function). One can check that the associated semi-infinite Hankel matrix
\begin{equation} H \ = \ \begin{pmatrix} 1 & 1 & 2 & 3 & 5 & \\ 1 & 2 & 3 & 5 & 8 & \\ 2 & 3 & 5 & 8 & 13 & \\ 3 & 5 & 8 & 13 & 21 & \\ 5 & 8 & 13 & 21 & 34 \\ & & & & & \ddots \end{pmatrix} \end{equation}
of the Fibonacci sequence is positive semi-definite, i.e. the
principal, finite minors of $H$ are all non-negative: It is enough
(in fact equivalent) to check that all $k \times k$ principal, contiguous minors are non-negative for $k \geq 1$.
Using the Fibonacci recurrence
and row operations the reader can verify
that the value of each $2 \times 2$ principal, contiguous minor
is one and that the value of each $k \times k$ principal, contiguous
minor is zero for $k \geq 3$.
In light of this positive semi-definiteness, a result of Alan Sokal (see theorem 2.1 of https://arxiv.org/pdf/1804.04498.pdf) indicates that the generating function $F(z)$ should have a presentation as a J-type continued fraction, i.e. there must exist a real number $\alpha_0 \geq 0$ along with two infinite sequences
\begin{equation} \begin{array}{ll} \displaystyle \underline{\beta} &\displaystyle = \ \big(\beta_1, \ \beta_2, \ \beta_3, \ \dots \big) \ \ \text{with $\beta_k \geq 0$ for all $k \geq 1$} \\ \displaystyle \underline{\gamma} &\displaystyle = \ \big(\gamma_0, \ \gamma_1, \ \gamma_2, \ \, \dots \big) \ \ \text{with $\gamma_k \in \Bbb{R}$ for all $k \geq 0$} \end{array} \end{equation}
such that
\begin{equation} F(z) \ = \ {\alpha_0 \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}
Question: Is anyone familiar with this continued fraction? What are $\alpha_0$, $\underline{\beta}$, and $\underline{\gamma}$ exactly?
thanks, ines.
Post-Script: I would point out that the J-continued fraction introduced here seems quite different from the continued fraction discussed in
