q-Catalan numbers are defined recurrently as C0=1, $C\_{N+1}=\sum\_{k=0}^N q^k C\_k C\_{N-k}$.


What can be said about the asymptotics of Cn when `0<q<1`?


P.S. In the case q>1 it is known that as n goes to infinity, $q^{-{n\choose 2}}C\_n(q)$ tends to the partition function $\prod\_{i=1}^\infty\frac1{1-q^{-i}}$. However, this doesn't help in the case `0<q<1`.



Asymptotics of q-Catalan numbers
Re Leonid's comment on a previous answer.


If the ratios $C\_{n+1}/C\_n$ converge, their limit $c(q)$ is such that $C(q,q/c(q))=c(q)$. 
Equivalently, $1/c(q)$ is the radius of convergence of the series $z\mapsto C(q,z)$. 
Or, writing $C(q,\cdot)$ as the ratio of two $q$-hypergeometric functions, one can show that $F(q,1/c(q))=0$, where 
$$
F(q,z)=\sum\_{n\ge0}(-1)^nq^{n^2-n}z^n/(q)\_n.
$$ 
This implies that $c(q)$ is the sum of a series in $q$ with integer coefficients, whose signs seem to be alternating starting with the coefficient of $q$. The first terms are
$$
c(q)=1+q+q^3-q^4+2q^5-3q^6+6q^7-12q^8+25q^9-52q^{10}+111q^{11}+\ldots
$$
The function $q\mapsto c(q)$ is nondecreasing on $q\ge0$, obvious values are $c(0)=1$ and $c(1)=4$, and as a holomorphic function, $c(\cdot)$ might have a pole inside the unit disk at about $q\approx-.4$. 
But apart from that...


