Determinantal formula for the largest-root distribution of non-central Wishart matrices

Let ${\mathcal R}_{n,m}(x,\lambda_1,\ldots,\lambda_m)$ denote the target function, and let $G_{n,m}(x,y)$ be defined recursively by

$$G_{n,2}(x,y)=n\,{}_0F_1(;n;xy)+y\,{}_0F_1(;n+1;xy)+(x-y-n+1)e^y\int_y^\infty e^{-t}{}_0F_1(;n+1;xt)\,dt,$$

and, for $m\geq 2$,

$$G_{n,m+1}(x,y)=\left(-y\frac{\partial^2}{\partial y^2}-(n-m+1)\frac{\partial}{\partial y}+x+m\right)G_{n,m}(x,y).$$

Determinantal formula. There is a function $C(x)$ such that

$${\mathcal R}_{n,m}(x,\lambda_1,\ldots,\lambda_m)=C(x)\det\left(\begin{array}{cc}{}_0F_1(;n-m+1;x\lambda_i)&\rangle_{j=1}\\[2pt]G_{n-m+j,j}(x,\lambda_i)&\rangle_{j=2}^m\end{array}\right),$$

where

$$C(x)=\frac{(n-m+1)x^{mn-\binom{m}{2}-1}e^{-mx}}{\prod_{k=1}^m(n-k+1)^k}.$$

The formula has been fully checked for $m=2$ and $m=3$, its front factor has been confirmed for $m=4$, and it is also true for $m=1$. Its significance is that it would exploit the factorization structure of the relevant holonomic system; the general formula beyond the checked cases remains conjectural.