It is possible to convert this infinite continued fraction
$\cfrac{1}{-a+\cfrac{b\;f(0)}{a+\cfrac{b\; f(1)}{-a+\cfrac{b\; f(2)}\ddots}}}$
to a special function ? Please, how do it?
where : $(a,b) >0$ and $f(n)=\cfrac{(n+1)^2}{4(n+1)^2-1}$ , $n \in N$, $f(0)=\cfrac{1}{3}$, $f(1)=\cfrac{4}{15}$, $f(2)=\cfrac{9}{35}$, ...