Hankel determinant of incomplete gamma functions

Question

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.

Answer 1

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.

Answer 2

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.