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

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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.

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2 Answers 2

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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.

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  • $\begingroup$ What if we consider the determinant of the entrywise square of this matrix. Can we still use Andreief Identity for it anyway? $\endgroup$
    – VSP
    Jun 2, 2021 at 13:53
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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.

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  • $\begingroup$ Did you want to post a comment on Marcel's answer instead? $\endgroup$ Apr 6, 2021 at 22:13
  • $\begingroup$ I re-cast the problem in terms of Laguerre polynomials, that appears to help. The objective is to study regimes where the asymptotics cannot be expected to be good. In engineering, the regime of interest is n~=r. There are still conditioning issues but the determinant is now diagonally dominant for tail probabilities (also the important regime) $\endgroup$
    – searp
    Apr 8, 2021 at 10:45

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