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
$$C(i+j,k+2)=\frac{k+2}{(2i+2j+k+2)}\binom{2i+2j+k+2}{i+...
- 83.4k
9
votes
Accepted
Some Hankel Determinants
After comparing Steven's original example with his new example, I believe I have an interesting generalization of both. Let $c \in \mathbb{C}$. Define the following three functions:
$$h(m)=\frac{1}{m-...
- 11.6k
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
$$ P(n,\alpha)=\frac{1}{r!}\int_0^\alpha e^{...
- 2,440
8
votes
Determinant of identity matrix plus Hilbert matrix
One thing you can say is that your determinant is the sum of determinants of the Cauchy matrices $C_S$ for subsets $S$ of $\{1,\ldots, n\}$, where $(C_S)_{i,j} = 1/(S_i + S_j - 1)$ for $1 \le i,j \le |...
- 52.6k
7
votes
Accepted
Hankel determinants of harmonic numbers
I prove your identity $$\sum_{j=0}^n (-1)^j\binom{n}{j}\binom{n+j}{j} H_j= 2(-1)^n H_{n}$$
which you claim to imply the result.
The method is the same as here.
At first, use $(-1)^k\binom{n+k}k=\...
- 94.7k
7
votes
Accepted
a Hankel matrix of involution numbers
As in arXiv:0902.1650 it suffices to show that $a(n,0)=I_n$ if $a(n,j)$ satisfies $a(n,j)=a(n-1,j-1)+a(n-1,j)+(j+1)a(n-1,j+1)$ with $a(n,-1)=0$ and $a(0,j)=[j=0]$.
But it is easily verified that $a(n,...
- 5,440
7
votes
Accepted
An identity related to Hankel determinants of $\sum_{k=1}^n \frac{2^k}{k}$
Define the sequence $a_n(x):=\sum_{j=0}^n(-2)^{n-j}\binom{n}j\binom{n+j}jx^j$ so that
$r(n)=\int_0^2\frac{a_n(x)-a_n(1)}{x-1}dx$.
We need the following fact which follows from the Vandermonde-Chu ...
- 40.2k
6
votes
Accepted
Moment problem on [-1,1]: necessary and sufficient conditions
If $\mu$ is a measure supported on $[-1,1]$, the change of variables $y = (1+x)/2$ gives you a measure $\rho$ supported on $[0,1]$, and the corresponding moments are related by
$$\int_0^1 y^k\; d\rho(...
- 52.6k
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 ...
- 2,359
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 $\log a$ ...
- 126k
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,...
- 1,563
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
- 51
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}...
- 40.2k
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 ...
- 126k
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 ...
- 5,440
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 ...
- 20.7k
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
$$P_n(x):=\...
- 40.2k
3
votes
Some nice polynomials related to Hankel determinants
This approach uses "Number Walls".
Given a sequence of elements
$\,a:\mathbb{Z}\to F\,$ in a field $\,F.\,$ Define the infinite matrix
of $\,n\times n\,$ Hankel determinants
$$ W_{n,m} :=\...
- 2,359
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 ...
- 40.2k
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 ...
- 28.1k
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 ...
- 21.5k
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 ...
- 40.2k
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 $...
- 13.3k
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 ...
- 1,241
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 ...
- 94.7k
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....
- 20.7k
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 + ...
- 28.7k
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}}} \...
- 2,359
Only top scored, non community-wiki answers of a minimum length are eligible