Hankel determinant of incomplete gamma functions I have some expressions that involve Hankel determinants of incomplete gamma functions.  They are of the ($r \times r$ form)

I'd like to evaluate these determinants.  Elementary operations help, but these determinants are so ill-conditioned (large $n$, $r$) that I have yet to find any technique (equilibration, etc) that provides sufficient stability.  There is a body of work  (KRATTENTHALER, Advanced Determinant Calculus) that casts the evaluation problem as a continued fraction, but actually working with these expressions is beyond my competence.  I am looking for any useful hints.
There is a more general problem of this type involving different functions, but if I can get some help understanding this problem, then I may be able to do the others.  Happy to share - this is old work that was never completed.
 A: Your quantity is
$$ P(n,\alpha)=\det_r A_{i,j}(n,\alpha),$$
with
$$ A_{i,j}(n,\alpha)=\int_0^\alpha t^{n+r-i-j}e^{-t}dt.$$
By the Andreief identity, this is
$$ P(n,\alpha)=\frac{1}{r!}\int_0^\alpha e^{-t_1}dt_1\cdots \int_0^\alpha e^{-t_r}dt_r \det_r(t_k^{n-i})\det_r(t_k^{r-j}),$$
and this is
$$ P(n,\alpha)=\frac{1}{r!}\int_0^\alpha e^{-t_1}dt_1\cdots \int_0^\alpha e^{-t_r}dt_r \prod_{k=1}^r t_k^{n-r}V(t)^2,$$
where $V(t)$ is the Vandermonde, $V(t)=\prod_{j<k}(t_k-t_j)$.
The above formulation shows that your quantity is basically the probability that all eigenvalues of a complex Wishart random matrix are smaller than $\alpha$. This should help you find its value.
For example, this was studied by Kurt Johansson in the paper Shape Fluctuations and Random Matrices, Commun. Math. Phys. 209, 437 – 476 (2000), and also by Iain M. Johnstone in the paper On the Distribution of the Largest Eigenvalue in Principal Components Analysis, The Annals of Statistics 29, 295-327 (2001).
For large $n$ and $r$, as functions of $\alpha$ it is related to the Tracy-Widom distribution.
A: Thanks!  I will follow up.  I re-cast the determinant as a better-conditioned one with entries that are partial inner products of Laguerre polynomials, I thought I'd try this first.
I am interested primarily in computing these things for n~=r and "large" r.  I am hoping to examine precisely the region where Tracy-Widom is unsatisfactory.
