TITLE: Determinant of a checkerboard Hankel matrix with Catalan numbers
QUESTION [12 upvotes]: My goal is to compute 
\begin{equation}
I =  \det \left(\mathbf{I} + \mathbf{A}\right)
\end{equation}
where $\mathbf{A}$ is a $n \times n$ checkerboard matrix filled with Catalan numbers:
$$
\left\{
  \begin{array}{crc}
    \mathbf{A}_{ij}  = C_{p-1} \alpha^{2(p-1)} &\mbox{ if }& i+j=2p \mbox{, and with }C_p= \frac{1}{p+1}\binom {2p} {p}\\
    \mathbf{A}_{ij}  =  0 &\mbox{ if }& i+j \mbox{ is odd.}
   \end{array}
\right.
$$
with $\alpha >0$ a parameter.
Numerically, it seems that $I$ has a limit when $n\to +\infty$ if $\alpha<1/2$ and diverges to $\infty$ otherwise.
Any idea ?

REPLY [5 votes]: If you consider the operator on functions that sends $f(x)$ to 
$$Tf(y) = \int_{-2}^2 \frac{ \sqrt{4-x^2} }{2\pi } \frac{1}{ 1- \alpha x y} f(\alpha x)  dx$$
then this is a well-defined integral operator from functions on $[-2\alpha,2\alpha]$ to functions on $[-2\alpha, 2\alpha]$ as long as $\alpha<1/2$. 
Moreover, if $f(x)=x^{i-1}$, then the coefficient of $y^{j-1}$ in $Tf(y)$ is exactly $\mathbf A_{ij}$. So the limit of your determinant should be exactly the Fredholm determinant of this integral operator.
I'm not sure if this helps at all.