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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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 ...
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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 ...
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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 ...