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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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### 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 ...
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### 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{...
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### 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 ...
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### 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 ...
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### 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_\...
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### 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 $$s_k = \int_{[-1,1]} x^k\,\mathrm{d}\mu,\quad \forall k\in\mathbb N\;?$$ ...
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### 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\...
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### 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\{ ...
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### 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} \...
### 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 ...