All Questions
Tagged with hankel-matrices determinants
13 questions
2
votes
0
answers
241
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,
$$A_{6,2}=\left ( \begin{matrix} 2 & 3 & 1& 0 &...
- 5,612
5
votes
0
answers
447
views
Determinant of Hankel matrix with $a_n=(n!)^2$
Consider a Hankel matrix of the form
$H_n(a_0(n))=\begin{pmatrix}
a_0(n) & (1!)^2 & (2!)^2 & \cdots & (n!)^2\\
(1!)^2 & (2!)^2 & (3!)^2& \cdots & ((n+1)!)^2\\
(2!)^2 &...
5
votes
1
answer
408
views
An interesting Hankel determinant
Let $h(n,t) = \sum\limits_{j = 0}^n {\binom
{\lfloor {\frac{n}{2}} \rfloor }{j}\binom
{\lfloor {\frac{n+1}{2}}\rfloor }{j}t^j \\ }.$
I am interested in the Hankel determinants $${D_k}(n,t) = \det \...
- 43.2k
16
votes
1
answer
896
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
$$
H_{n}:=\begin{pmatrix}
h_{0} & h_{1} & \dots & h_{n-1}\\
h_{1} & ...
- 17k
4
votes
2
answers
242
views
Hankel determinant of incomplete gamma functions
I have some expressions that involve Hankel determinants of incomplete gamma functions. They are of the ($r \times r$ form)
I'd like to evaluate these determinants. Elementary operations help, but ...
- 41
12
votes
2
answers
2k
views
Determinant of identity matrix plus Hilbert matrix
I am looking for the determinant
$$ \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{...
- 51
5
votes
1
answer
276
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\...
- 1,271
13
votes
1
answer
385
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 ...
- 1,593
5
votes
2
answers
635
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
$$\lim_{q\to1}\frac{d(n)}{(q-1)^\binom{n+...
- 977
10
votes
4
answers
554
views
Hankel determinants of harmonic numbers
Let $H_n=\sum_{k=1}^n\frac 1 k$ be the $n$-th harmonic number with $H_0=0.$
Question: Is the following true?
$$\det\left(H_{i+j}\right)_{i,j=0}^n=(-1)^n \frac{2H_{n}}{n! \prod_{j=1}^n \binom{2j}{j} \...
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 ...
- 43.2k
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 ...
- 43.2k
12
votes
2
answers
779
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\{
...
- 148k