# Questions tagged [hankel-matrices]

Hankel matrices are square matrices whose coefficients $a_{i,j}$ only depend on the sum $i+j$. They appear in particular in the theory of orthogonal polynomials on the real line.

22
questions

**12**

votes

**1**answer

359 views

### A matrix identity related to Catalan numbers

Let $$C_n=\frac{1}{2n+1}\binom{2n+1}{n}$$ be a Catalan number. It is well-known that $$(\sum_{n\ge{0}}C_n x^n)^k=\sum_{n\ge{0}}C(n,k)x^n$$ with $$C(n,k)=\frac{k}{2n+k}\binom{2n+k}{n}.$$
It is also ...

**12**

votes

**1**answer

310 views

### Some more binomial coefficient determinants

The setup is similar to this question, but generalizes the size of the Hankel matrix. We'll define
$$d(n,k,r):=\det\left(\binom{2i+2j+k+r}{i+j}\right)_{i,j=0}^{kn-1}.$$
Edit: Thanks to Johann ...

**3**

votes

**1**answer

157 views

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

Let ${\left( {{a_n}} \right)_{n \geqslant 0}}$ be a sequence of real numbers and ${H_n} = \det \left( {{a_{i + j}}} \right)_{i,j = 0}^{n - 1}$ the $n-$th Hankel determinant. The sequence ${\left( {{...

**2**

votes

**0**answers

418 views

### Eigenvalues of a specific Hankel matrix

I have an $\frac{N}{2} \times \frac{N}{2}$ matrix $G$ with entries given by
\begin{equation}
G_{ij} = \frac{1}{\sin(\frac{\pi}{N}(i+j-\frac{3}{2}))}, \;\;\;\;\;\;\;\; 1 \le i,j \le \frac{N}{2},
\end{...

**12**

votes

**2**answers

1k views

### Determinant of identity matrix plus Hilbert matrix

I am looking for the determinant
$$ \det(I_n + H_n) $$
where $I_n$ is the $n \times n$ identity matrix and $H_n$ is the $n \times n$ Hilbert matrix, whose entries are given by
$$ [H_n]_{ij} = \frac{...

**1**

vote

**0**answers

78 views

### Properties of graphs with Hankel-like adjacency matrix

I am having undirected graphs with adjacency matrices which have a regular Hankel-like form, e.g.,
$$A=\begin{pmatrix}0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & (6\times 0 \text{ ...

**9**

votes

**0**answers

166 views

### Some quotients of Hankel determinants

This question has been inspired by Hankel determinants of binomial coefficients.
For a sequence $\{h_{n}\}_{n=0}^{\infty}$ denote by $H_n$ the Hankel matrix
$$H_{n}:=\begin{pmatrix}
h_{0} & h_{...

**4**

votes

**1**answer

195 views

### Hankel determinant evaluation of special lattice paths

Let $n$ be a positive integers and $T=T_{n,n}$ be the $n\times n$ table in the first quadrant composed of $n^2$ unit squares, whose $(x,y)$-blank is locate in the $x^{th}$-column ...

**5**

votes

**2**answers

534 views

### Some curious Hankel determinants

Let $f(n,q)=\prod_{j=1}^na(q^j)$ for a polynomial $a(q)$ and let $d(n)=\det(f(i+j,q))_{i,j=0}^n$ be its Hankel determinant.
Computer experiments suggest that
$$\lim_{q\to1}\frac{d(n)}{(q-1)^\binom{n+...

**5**

votes

**1**answer

440 views

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

This question is related to Hankel determinants of harmonic numbers.
Let $f(n)=\sum_{k=1}^n \frac{2^k}{k}$ and $r(n)=\sum_{j=0}^n (-2)^{n-j}\binom{n}{j}\binom{n+j}{j}f(j).$
In order to compute the ...

**10**

votes

**4**answers

461 views

### Hankel determinants of harmonic numbers

Let $H_n=\sum_{k=1}^n\frac 1 k$ be the $n$-th harmonic number with $H_0=0.$
Question: Is the following true?
$$\det\left(H_{i+j}\right)_{i,j=0}^n=(-1)^n \frac{2H_{n}}{n! \prod_{j=1}^n \binom{2j}{j} \...

**11**

votes

**1**answer

296 views

### a Hankel matrix of involution numbers

Let $I_k$ denote the enumeration of involutions among permutations in $\mathfrak{S}_k$. I always enjoy these numbers. Of course, here is yet another cute experimental finding for which I ask validity. ...

**2**

votes

**0**answers

121 views

### Sufficient condition for a solution to Hamburger moment problem

Let $\{m_n\}_{n=0}^{\infty}$ be a sequence of real numbers.
It is well known that there exist a positive Borel measure $\mu$ on the real line with moments given by $\{m_n\}_{n=0}^{\infty}$ if and ...

**12**

votes

**1**answer

450 views

### Hankel determinants of binomial coefficients

For $\{h_{n}\}_{n=0}^{\infty}$ a real sequence, detone by $H_{n}$ the $n\times n$ Hankel matrix of the form
$$
H_{n}:=\begin{pmatrix}
h_{0} & h_{1} & \dots & h_{n-1}\\
h_{1} & h_{2} &...

**8**

votes

**4**answers

1k views

### Some Hankel Determinants

After invoking a recursion relation for Hankel determinants in my answer to a (mostly unrelated) question, I started wondering what else I could use this recursion for, and stumbled upon some results ...

**5**

votes

**1**answer

294 views

### Eigenvalues of partial Hankel matrices

I was wondering if there are closed formulas for the singularvalues of a partial Hankel matrix (by partial I mean $\ell<n$)
\begin{align*}
H=
\begin{bmatrix}
c_1 & c_2 & \ldots & c_\...

**4**

votes

**2**answers

313 views

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

Consider a sequence of real numbers $s=(s_0,s_1,\ldots)$. When is there a Borel measure $\mu$ supported on $[-1,1]$ so that
$$
s_k = \int_{[-1,1]} x^k\,\mathrm{d}\mu,\quad \forall k\in\mathbb N\;?
$$
...

**5**

votes

**1**answer

204 views

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

Consider the sequence $(\varphi_i)$ of reverse Bessel polynomials which begins as follows.
\begin{align*}
\varphi_0&=1\\
\varphi_1&=x\\
\varphi_2&=x^2 + x\\
\varphi_3&=x^3 + 3x^2 + 3x\...

**12**

votes

**2**answers

551 views

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

My goal is to compute
\begin{equation}
I = \det \left(\mathbf{I} + \mathbf{A}\right)
\end{equation}
where $\mathbf{A}$ is a $n \times n$ checkerboard matrix filled with Catalan numbers:
$$
\left\{
...

**4**

votes

**0**answers

316 views

### When is a Hankel matrix invertible?

I would like to have some conditions that render a general Hankel matrix of the form
\begin{pmatrix}a_1 & a_2 & a_3 &\cdots & a_n \\ a_2 & a_3 & a_4 & \cdots & a_{n+1} \...

**6**

votes

**1**answer

248 views

### Diagonalization of the matrix $(1/(i+j+\rm{const}))_{i,j}$

Consider the following infinite matrix: $A_{i,j}=\frac1{i+j+\gamma}$, $0\leq i,j<\infty$, $\gamma>0$ is a constant. Is it known how to diagonalize $A$, or, say, calculate $(I+tA)^{-1}$ for ...

**5**

votes

**0**answers

461 views

### A class of determinants associated to Catalan-like Hankel determinants

The following matrices are related to some Catalan-like Hankel matrices. My question is whether direct computations of determinants of such matrices (i.e. without recourse to Hankel determinants) ...