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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.

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Bijective proof of deteminant formula for Hankel matrix of central binomial coefficients

Is there a nice bijective proof of the fact that the determinant of the $(n+1)$-by-$(n+1)$ Hankel matrix whose respective entries are the central binomial coefficients $0 \choose 0$, $2 \choose 1$, $\...
James Propp's user avatar
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1 vote
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Can all real positive semidefinite Hankel matrices be decomposed into sum of rank 1 real positive semidefinite Hankel matrices?

Denote the set of real positive semidefinite $d\times d$ Hankel matrices as $\mathcal{S}$. Can we always decompose one $S\in \mathcal{S}$ into sum of rank $1$ $S_i\in \mathcal{S}$, i.e., $S=\sum_i S_i$...
zzy's user avatar
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2 votes
0 answers
237 views

Determinants of band matrices which are related to Hankel matrices of Catalan numbers

Let $A_{n,m}$ be the band matrix $$ A_{n,m}=\left( \binom{2m}{j-i+m}-\binom{2m}{m-i-j-1} \right)_{0\leq {i,j} \leq {n-1}}.$$ For example, $$A_{6,2}=\left ( \begin{matrix} 2 & 3 & 1& 0 &...
Johann Cigler's user avatar
1 vote
0 answers
78 views

Shifted Hankel determinants for convolutions of Catalan numbers

It is well known that for $m\in \mathbb N$ the Hankel determinants $$D_m(n)= \det\left(C_{i+j+m}\right)_{0\leq i,j\leq {n-1}}$$ satisfy $D_m(n)=p_m(n)$, where $p_m(n)=\prod_{1 \leq i \leq j \leq {m-1}}...
Johann Cigler's user avatar
7 votes
0 answers
249 views

Hankel determinants for some convolutions of Catalan numbers

Let $c(x)=\frac{1-\sqrt{1-4x}}{2x}$ be the generating function of the Catalan numbers and let $$x^k c(x)^{2k}=(c(x)-1)^k =\sum_{n\geq0}c(k,n)x^n.$$ Consider the determinants $$D(k,n,m)= \det\left(c(k,...
Johann Cigler's user avatar
5 votes
0 answers
427 views

Determinant of Hankel matrix with $a_n=(n!)^2$

Consider a Hankel matrix of the form $H_n(a_0(n))=\begin{pmatrix} a_0(n) & (1!)^2 & (2!)^2 & \cdots & (n!)^2\\ (1!)^2 & (2!)^2 & (3!)^2& \cdots & ((n+1)!)^2\\ (2!)^2 &...
fs98's user avatar
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1 vote
1 answer
181 views

A continued J fraction for $a_n = \frac{1}{(n+1)^2}$?

The following is called a J continued fraction: $$\cfrac{\alpha_0}{1+a_0x-\cfrac{b_1x^2}{1+a_1x-\cfrac{b_2x^2}{1+a_2x-\cdots}}}$$ where the constants are real numbers. Let $\alpha_n= \frac{1}{(n+1)^2}$...
VSP's user avatar
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A question on continued J-fraction

Consider the following two continued fractions $A$ and $B$: $$\frac{\alpha_0}{1+a_0x-\frac{b_1x^2}{1+a_1x-\frac{b_2x^2}{1+a_2x-\cdots}}}$$ $$\frac{\beta_0}{1+c_0x-\frac{d_1x}{1+c_1x-\frac{d_2x}{1+c_2x-...
VSP's user avatar
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5 votes
1 answer
394 views

An interesting Hankel determinant

Let $h(n,t) = \sum\limits_{j = 0}^n {\binom {\lfloor {\frac{n}{2}} \rfloor }{j}\binom {\lfloor {\frac{n+1}{2}}\rfloor }{j}t^j \\ }.$ I am interested in the Hankel determinants $${D_k}(n,t) = \det \...
Johann Cigler's user avatar
3 votes
0 answers
126 views

Positivity of sequences

Totally positive sequences $\lbrace a_n\rbrace_{n\in\mathbb{Z}}$ are defined as those such that the Töplitz matrix $A_{ij}=a_{i-j}$ is totally positive (all its minors are non-negative). An ...
Nicolas Medina Sanchez's user avatar
2 votes
2 answers
480 views

Some nice polynomials related to Hankel determinants

Let $f_n(x)=\prod_{j=0}^{\lfloor{\frac{n-1}{2}}\rfloor}\prod_{i=2j+1}^{2n-2j-1}\frac{2x+i}{i}$ and $g_n(x)=\prod_{j=1}^{\lfloor{\frac{n}{2}}\rfloor}\prod_{i=2j}^{2n-2j}\frac{2x+i}{i}.$ Then $f_n(k)=\...
Johann Cigler's user avatar
4 votes
2 answers
208 views

Hankel determinant of incomplete gamma functions

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 ...
searp's user avatar
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Function of several variables whose hessian is a Hankel matrix

First of all, let me apologize because I asked this question a few days ago on https://math.stackexchange.com, but I did not get any reply. I am studying a function $f:\mathbb{R}^n\rightarrow\mathbb{R}...
G. Fougeron's user avatar
2 votes
2 answers
362 views

Number of bounded Dyck paths with negative length as Hankel determinants

This is a continuation of my post Number of bounded Dyck paths with "negative length". Let $C_{n}^{(2k+1)}$ be the number of Dyck paths of semilength $n$ bounded by $2k+1.$ They satisfy a ...
Johann Cigler's user avatar
12 votes
1 answer
544 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 ...
Johann Cigler's user avatar
13 votes
1 answer
379 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 ...
Wolfgang's user avatar
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3 votes
1 answer
183 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( {{...
Johann Cigler's user avatar
2 votes
0 answers
539 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{...
Marc's user avatar
  • 101
12 votes
2 answers
2k 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{...
Tobi's user avatar
  • 121
1 vote
0 answers
143 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{ ...
pisoir's user avatar
  • 243
9 votes
0 answers
211 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_{...
Johann Cigler's user avatar
4 votes
1 answer
243 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 ...
d.y's user avatar
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5 votes
2 answers
628 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+...
Johann Cigler's user avatar
5 votes
1 answer
496 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 ...
Johann Cigler's user avatar
10 votes
4 answers
552 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} \...
Johann Cigler's user avatar
11 votes
1 answer
330 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. ...
T. Amdeberhan's user avatar
2 votes
0 answers
151 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 ...
Boby's user avatar
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16 votes
1 answer
857 views

Hankel determinants of binomial coefficients

For $\{h_{n}\}_{n=0}^{\infty}$ a real sequence, denote 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} & ...
Twi's user avatar
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8 votes
4 answers
3k 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 ...
Steven Landsburg's user avatar
6 votes
1 answer
388 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_\...
Anahita's user avatar
  • 363
6 votes
2 answers
426 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 $$ s_k = \int_{[-1,1]} x^k\,\mathrm{d}\mu,\quad \forall k\in\mathbb N\;? $$ ...
MERTON's user avatar
  • 505
5 votes
1 answer
271 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\...
Simon Willerton's user avatar
12 votes
2 answers
755 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\{ ...
user16215's user avatar
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5 votes
0 answers
419 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} \...
baronbrixius's user avatar
6 votes
1 answer
273 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 ...
Fedor Petrov's user avatar
5 votes
0 answers
482 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) ...
Johann Cigler's user avatar