12
votes

Accepted

### A matrix identity related to Catalan numbers

After unpacking the equation $$\left( {{C(i+j,k+2)}} \right)_{i,j = 0}^{n - 1}=A_{n}G_{n,k} A_{n}^T$$
we see that we want to prove the identity
$$C(i+j,k+2)=\frac{k+2}{(2i+2j+k+2)}\binom{2i+2j+k+2}{i+...

9
votes

Accepted

### Some Hankel Determinants

After comparing Steven's original example with his new example, I believe I have an interesting generalization of both. Let $c \in \mathbb{C}$. Define the following three functions:
$$h(m)=\frac{1}{m-...

8
votes

### Hankel determinant of incomplete gamma functions

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^{...

8
votes

### Determinant of identity matrix plus Hilbert matrix

One thing you can say is that your determinant is the sum of determinants of the Cauchy matrices $C_S$ for subsets $S$ of $\{1,\ldots, n\}$, where $(C_S)_{i,j} = 1/(S_i + S_j - 1)$ for $1 \le i,j \le |...

7
votes

Accepted

### Hankel determinants of harmonic numbers

I prove your identity $$\sum_{j=0}^n (-1)^j\binom{n}{j}\binom{n+j}{j} H_j= 2(-1)^n H_{n}$$
which you claim to imply the result.
The method is the same as here.
At first, use $(-1)^k\binom{n+k}k=\...

7
votes

Accepted

### a Hankel matrix of involution numbers

As in arXiv:0902.1650 it suffices to show that $a(n,0)=I_n$ if $a(n,j)$ satisfies $a(n,j)=a(n-1,j-1)+a(n-1,j)+(j+1)a(n-1,j+1)$ with $a(n,-1)=0$ and $a(0,j)=[j=0]$.
But it is easily verified that $a(n,...

7
votes

Accepted

### An identity related to Hankel determinants of $\sum_{k=1}^n \frac{2^k}{k}$

Define the sequence $a_n(x):=\sum_{j=0}^n(-2)^{n-j}\binom{n}j\binom{n+j}jx^j$ so that
$r(n)=\int_0^2\frac{a_n(x)-a_n(1)}{x-1}dx$.
We need the following fact which follows from the Vandermonde-Chu ...

6
votes

Accepted

### Moment problem on [-1,1]: necessary and sufficient conditions

If $\mu$ is a measure supported on $[-1,1]$, the change of variables $y = (1+x)/2$ gives you a measure $\rho$ supported on $[0,1]$, and the corresponding moments are related by
$$\int_0^1 y^k\; d\rho(...

6
votes

Accepted

### Hankel determinant evaluation of special lattice paths

The $d(n):=D(n,n)$ is OEIS sequence A005773. The Hankel determinant property is given in the sequence entry. Also the recursion $\;nd_{n}=2nd_{n-1}+3(n-2)d_{n-2}$. A proof could come from a similar ...

5
votes

### Some curious Hankel determinants

The identity is clearly invariant to scaling $a$ by a constant. (One has to check the constant appears in the determinant to power $n(n+1)$). So wlog we may assume $a(1)=1$.
We can expand $\log a$ ...

5
votes

Accepted

### Some more binomial coefficient determinants

Johann Cigler and I have posted a proof of many of these observations on arXiv:
"An interesting class of Hankel determinants", arXiv:1807.08330.
Let $d_r(N)=\det\left({2i+2j+r\choose i+j}\right)_{i,...

5
votes

### Determinant of identity matrix plus Hilbert matrix

If you look at $H=\frac 1{(i+j-1)\pi}$ instead, then $\det(1 + H)\sim n^{3/8}$, as $n\to\infty$. This is basically proven in arXiv:1808.08009, arXiv:1905.03154

5
votes

Accepted

### An interesting Hankel determinant

Denote $a_n=a(n,t)$ and $b_n=b(n,t)$. To help avoiding the min function, write
$$b_n=\binom{n+3}3t^n+\sum_{j=0}^{n-1}\binom{3+j}3\left[t^{2n-j}+t^j\right].$$
Notice that $a_n=\frac{nt^{n+1}-(n+1)t^n+1}...

5
votes

### Determinant of a checkerboard Hankel matrix with Catalan numbers

If you consider the operator on functions that sends $f(x)$ to
$$Tf(y) = \int_{-2}^2 \frac{ \sqrt{4-x^2} }{2\pi } \frac{1}{ 1- \alpha x y} f(\alpha x) dx$$
then this is a well-defined integral ...

4
votes

### Hankel determinants of harmonic numbers

As asked by Fedor Petrov I sketch the missing details.
If $a(n)$ is any sequence with $a(0)=1$, such that all Hankel determinants $M_n=\det\left(a(i+j)\right)_{i,j=0}^n$ are $ \neq 0$, define a ...

4
votes

Accepted

### Some nice polynomials related to Hankel determinants

See Example 4 in Section 3.1.6 of https://arxiv.org/abs/1409.2562 (Federico Ardila, "Algebraic and geometric methods in enumerative combinatorics").
It explains that your $g_n(k)$ has an ...

4
votes

3
votes

### Hankel determinants of harmonic numbers

We propose a proof (somewhat different from Fedor's) for the crucial relation
$$\sum_{j=0}^n (-1)^j\binom{n}{j}\binom{n+j}{j} H_j= 2(-1)^n H_{n}.\tag1$$
To this end, define the polynomials
$$P_n(x):=\...

3
votes

### Some nice polynomials related to Hankel determinants

This approach uses "Number Walls".
Given a sequence of elements
$\,a:\mathbb{Z}\to F\,$ in a field $\,F.\,$ Define the infinite matrix
of $\,n\times n\,$ Hankel determinants
$$ W_{n,m} :=\...

3
votes

### Hankel determinants of binomial coefficients

This is not an answer.
Although there might be no "closed formula" for the case $p=4$, I wish to add the following to the case $p=2$ for which the determinants evaluate to $2^{n-1}$.
Suppose we ...

3
votes

### Moment problem on [-1,1]: necessary and sufficient conditions

The following paper answers a more general version of this question, namely the Hausdorff moment problem for compact sets in $\mathbb{R}^n$.
This is a fairly broad topic, and you may enjoy learning ...

2
votes

### Some Hankel Determinants

Not yet a fully satisfying answer, but Theorem 1 does generalize.
Given two sequences $j$ and $k$, define the sequence $h$ by
$$h(m)=j(m-1)k(m)$$
Let $H$, $J$ and $K$ be the corresponding families ...

2
votes

### Some Hankel Determinants

In as far as the computation goes, these determinants all fall out easily from the method shown in this paper.
When inductive proofs work, this technique is simpler to use than any other listed in ...

2
votes

### Eigenvalues of partial Hankel matrices

Typically, one does "Prony method": considers an infinite (or just long enough) sequence $c=(c_1,c_2,\dots)$ and a system of equations of the form $V(x)Z=c$, with $Z$ a vector of non-0 unknowns, and $...

2
votes

Accepted

### Evaluation of Hankel determinants for the reverse Bessel polynomials

There is a combinatorial formula for the Hankel determinant $H_n= \det\left([\varphi_{i+j}]_{i,j=0}^{n}\right)$ in terms of weighted sums of disjoint collections of Schröder paths (and also for ...

2
votes

### Determinant of a checkerboard Hankel matrix with Catalan numbers

Since $$\binom{2p}{p} =O\left(\frac{4^p}{\sqrt{p}}\right),$$ the coefficients of your matrix are unbounded when $\alpha > 1/2,$ while for $\alpha < 1/2$ the matrix $A$ is well-approximated by ...

2
votes

### Hankel determinants of harmonic numbers

Identities involving harmonic numbers that are of interest for physicists, Utilitas Mathematica 83 (2010), 291-299, H. Prodinger.
This paper contains the identity (1) as well.
Now starts Johann ...

2
votes

Accepted

### Number of bounded Dyck paths with negative length as Hankel determinants

Here's how I think this can be proved based on what Richard Stanley already did in your previous question.
If we take the network in Section 3.1.6, Example 4 part (a) of https://arxiv.org/abs/1409....

1
vote

### Number of bounded Dyck paths with negative length as Hankel determinants

As stated in the previous question, $C_n^{(2k+1)}$ satisfies
$$\sum_{j=0}^{k+1} (-1)^j \binom{2k+2-j}{j} C_{n-j}^{(2k+1)}=0.$$
The formula
$$C_{ - n}^{(2k + 1)} = \det \left( {C_{n + 1 + i + j}^{(2k + ...

1
vote

### The Golay-Rudin-Shapiro sequence as “Hankel transform”

Here is an attempt at analyzing this problem. It seems better to adjust the index of the sequence, so let $\;A_n = b_k \mbox{ if } n=2^k,\; 0\;$ otherwise. Let $\;c_n\; := \det \left( {{A_{i + j}}} \...

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