called multiple Laguerre polynomials of type I (with respect to the weights $x^{M-N}e^{-Mx}$ and $x^{M-N}e^{-Ma^{-1}x}$) [@Ap1], [@Ap2]. These polynomials exist and are unique. Moreover, they admit integral expressions [@BK1].

Let us define the constants $h^{(1)}_{n_1,n_2}$ and $h^{(2)}_{n_1,n_2}$ to be $$\label{eq:normalize}
\begin{split}
h^{(1)}_{n_1,n_2}&=\int_{0}^{\infty}L_{n_1,n_2}(x)x^{n_1+M-N}e^{-Mx}dx,\\
h^{(2)}_{n_1,n_2}&=\int_{0}^{\infty}L_{n_1,n_2}(x)x^{n_2+M-N}e^{-Ma^{-1}x}dx.
\end{split}$$

Then the following result in [@BK1] and [@DF] allows us to express the correlation kernel in (\[eq:mpoint\]) in terms of a finite sum of the multiple Laguerre polynomials.

\[pro:CD\] The correlation kernel in $K_{M,N}(x,y)$ (\[eq:mpoint\]) can be expressed in terms of multiple Laguerre polynomials as follows $$\label{eq:ker}
\begin{split}
\left(xy\right)^{\frac{N-M}{2}}(x-y)K_{M,N}(x,y)&=L_{N_0,N_1}(x)Q_{N_0,N_1}(x)\\
&-\frac{h^{(1)}_{N_0,N_1}}{h^{(1)}_{N_0-1,N_1}}L_{N_0-1,N_1}(x)Q_{N_0+1,N_1}(x)\\
&-\frac{h^{(2)}_{N_0,N_1}}{h^{(2)}_{N_0,N_1-1}}L_{N_0,N_1-1}(x)Q_{N_0,N_1+1}(x)
\end{split}$$ where $N_0=N-N_1$.

This result allows us to compute the limiting kernel once we obtain the asymptotics for the multiple Laguerre polynomials.

Stieltjes transform of the eigenvalue distribution {#se:Stie}
==================================================

In order to study the asymptotics of the correlation kernel, we would need to know the asymptotic eigenvalue distribution of the Wishart ensemble (\[eq:wishart\]). Let $F_N(x)$ be the empirical distribution function (e.d.f) of the eigenvalues of $B_N$ (\[eq:BN\]). The asymptotic properties of $F_N(x)$ can be found by studying its Stieltjes transform.

The Stieltjes transform of a probability distribution function (p.d.f) $G(x)$ is defined by $$\label{eq:stie}
m_G(z)=\int_{-\infty}^{\infty}\frac{1}{\lambda-z}dG(x), \quad
z\in\mathbb{C}^+=\left\{z\in\mathbb{C}:\mathrm{Im}(z)>0\right\}.$$ Given the Stieljes transform, the p.d.f can 