linear algebra – Invertibility of special Hankel matrix

At first the simple case. Let $h_kin mathbb C$ of the form
$$h_k=sum_{j=1}^Mc_j;z_j^kqquadqquad text{for }kinmathbb Z_{>0}.$$
Where $c_jin mathbb C$ are some weights and $z_jin mathbb C$ pairwise distinct with $|z_j|=1$. The normalize condition should not be that important, but maybe this additional property is getting useful for my main question. Now lets define the Hankel matrix $H(s)$ for $sin mathbb N_0$ given by
$$H(s)=(h_{s+m+l})_{m=0,l=0}^{M-1,M-1}=begin{pmatrix}h_{s}&dots&h_{M-1+s}\vdots&&vdots\h_{M-1+s}&dots&h_{2M-2+s}end{pmatrix}.$$
We can now find a decomposition of this matrix by
begin{align*} H(s)&=begin{pmatrix}z_1^{s}&dots&z_M^s\vdots&&vdots&\z_1^{M-1+s}&dots&z_M^{M-1+s}end{pmatrix}
begin{pmatrix} c_1&&\&ddots\&&c_Mend{pmatrix}
begin{pmatrix}1&dots&z_1^{M-1}\vdots&&vdots&\1&dots&z_M^{M-1}end{pmatrix}.end{align*}

We can than look on the 3 different determinants and will see, that $H(s)$ is indeed invertible. Ok so far so good.

Now to my main question. We will now look on $h_k$ defined by
$$h_k=sum_{j=1}^Mphi_j(k);z_j^k.$$
Where $phi_j$ are some complex polynomials of degree $d$ and $z_j$ as before. Again we are looking on the Hankel matrix
$$h(s)=(h_{s+m+l})^{(d+1)cdot(M-1)}_{m,l=0}.$$
So it is the ‘same’ matrix, but now with new coefficients and bigger.
I assume that kind of matrix to be invertible since, I am calculating a lot with them and they was always (even numerical) invertible. Furthermore I found some remarks where they said they should be invertible. How can I show, that this new Hankel matrix is still invertible? Is there an similar decomposition like the simple case before?
This problem is strongly related with Prony’s Method (maybe it will help you).

reference request – Some nice polynomials related to Hankel determinants

Let
$f_n(x)=prod_{j=0}^{n}prod_{i=2j+1}^{2n-2j-1}frac{2x+i}{i}$
and
$g_n(x)=prod_{j=1}^{n-1}prod_{i=2j}^{2n-2j}frac{2x+i}{i}.$

Then
$f_n(k)=det left( {f_{n+i+j}(1) } right)_{i,j = 0}^{k – 1}$ for each positive integer $k$ and analogously for $g_n(k).$

Note that $f_n(1)=binom{2n+1}{n}$ and $g_n(1)=C_n,$ a Catalan number.

Do these polynomials also occur in other contexts? Are there other integer sequences $a_n$

such that the polynomials $p_n(x)$ with $p_n(k)=det left( {a_{n+i+j} } right)_{i,j = 0}^{k – 1}$ are nice?

Hankel determinant of incomplete gamma functions

I have some expressions that involve Hankel determinants of incomplete gamma functions. The are of the (rxr form) enter image description here

My problem is that 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.

complete asymptotic series for Hankel function

How the complete asymptotic series of Hankel’s function as $xrightarrowinfty$ is given by $H^{(1)}_p (x) sim sqrt {frac{2}{pi x}} e^{(+ix’)} sum_{n=0}^{infty} a_n x^{-n}$ (while $x’=x-frac{ppi}{2}-frac{pi}{4}$) ??

If $W(x)=sum_{n=0}^{infty} a_n x^{-n alpha-frac{1}{2}}$ with

$a_0=1$,
$a_n= frac {(-i)^n}{2^n n!} prod_{k=1}^{n} {(k-frac {1}{2})^2 – p^2)}, (ngeq1)$.

With the ODE equation $x^2W”+(2ix^2+x)W’+(ix-p^2)W=0$.

plotting – Can’t plot derivative of Hankel function

I am trying to plot the first derivative of $H^{(1)}_3 (ix)$ with respect to its argument $ix$, where $H^{(1)}_3 (ix)$ is the Hankel function of the first kind with order $3$, $x in mathbb{R}$ is a real variable and $i = sqrt{-1}$ is the imaginary unit.

According to the chain rule, it should be:

$$frac{mathrm{d}}{mathrm{d}(ix)} left( H^{(1)}_3 (ix) right) = frac{1}{i} frac{mathrm{d}}{mathrm{d}x} left( H^{(1)}_3 (ix) right)$$

So, first, I defined:

a(x_) := HankelH1(3, I*x);

It is a real-valued function and it’s easy to plot, for example with Plot(a(X), {X, 0, 10}).

Then:

b(x_) := D(a(x), x);

which is pure imaginary and should represent $displaystyle frac{mathrm{d}}{mathrm{d}x} left( H^{(1)}_3 (ix) right)$. Then,

c(x_) := D(a(x), x) / I;

should be $displaystyle frac{mathrm{d}}{mathrm{d}(ix)} left( H^{(1)}_3 (ix) right)$ and should be real.

However, I can’t plot it:

Plot(c(X), {X, 0, 10})

gives

General::ivar: 0.0002042857142857143` is not a valid variable.
General::ivar: 0.20428591836734694` is not a valid variable.
General::ivar: 0.40836755102040817` is not a valid variable.
General::stop: Further output of General::ivar will be suppressed during this calculation.

Why? What am I doing wrong?

I’m using Mathematica 12.0.0 on Linux x86 (64 bit).