Related papers: An interesting class of Hankel determinants

In this note, we present the determinant, the inverse and a lower bound for the smallest eigenvalue for some Hankel matrices

We study some polynomials which are related to Hankel determinants of backward shifts of the coefficients of a partial theta function. In this version an appendix is added which gives a simple formula for the coefficients of the reciprocal…

Michael Somos conjectured a relation between Hankel determinants whose entries $\frac 1{2n+1}\binom{3n}n$ count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by…

For any field k and any integers m,n with 0 <= 2m <= n+1, let W_n be the k-vector space of sequences (x_0,...,x_n), and let H_m be the subset of W_n consisting of the sequences that satisfy a degree-m linear recursion, that is, for which…

We investigate the simplest class of hyperdeterminants defined by Cayley in the case of Hankel hypermatrices (tensors of the form $A_{i_1i_2... i_k}=f(i_1+i_2+...+i_k)$). It is found that many classical properties of Hankel determinants can…

We work with detail the Drinfeld module over the ring $$A=F_2[x,y]/(y^2+y=x^3+x+1).$$ The example in question is one of the four examples that come from quadratic imaginary fields with class number $h = 1$ and rank one. We develop specific…

Fix $n$ a positive integer. Take the $n$-th metallic number $\phi_n=\frac{n+\sqrt{n^2+4}}{2}$ (e.g. $\phi_1$ is the golden number) and let $\Phi_n(q)$ be its $q$-deformation in the sense of S. Morier-Genoud and V. Ovsienko. This is an…

We show that the Hankel determinants of a generalized Catalan sequence satisfy the equations of the elliptic sequence. As a consequence, the coordinates of the multiples of an arbitrary point on the elliptic curve are expressed by the…

In this paper we characterise the indeterminate case by the eigenvalues of the Hankel matrices being bounded below by a strictly positive constant. An explicit lower bound is given in terms of the orthonormal polynomials and we find…

The n-th Delannoy number and the n-th Schr\"oder number given by $D_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}$ and $S_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}/(k+1)$ respectively arise naturally from enumerative combinatorics. Let p be an…

This (partly expository) paper originated from the study of Hankel determinants of convolution powers of Catalan numbers and of Narayana polynomials. This led to some Hankel determinants of signed Catalan numbers whose values are multiples…

We prove there exist infinitely many odd integers $n$ for which there exists a pair of positive divisors $d_1, d_2>1$ of $(n^2+1)/2$ such that $$d_1+d_2=\delta n+(\delta+2).$$ We prove the similar result for $\varepsilon=\delta-2$ and…

We calculate the effect of simple Hecke operators on u-expansions of higher rank Drinfeld modular forms, the eigenvalue for the Drinfeld discriminant function $\Delta_t$ and show that a certain natural class of Hecke operators is completely…

The estimates for the second Hankel determinant a_2a_4-a_3^2 of analytic function f(z)=z+a_2 z^2+a_3 z^3+...b for which either zf'(z)/f(z) or 1+zf"(z)/f'(z) is subordinate to certain analytic function are investigated. The estimates for the…

Let $O(2n+\ell)$ be the group of orthogonal matrices of size $\left(2n+\ell\right)\times \left(2n+\ell\right)$ equipped with the probability distribution given by normalized Haar measure. We study the probability \begin{equation*}…

I give simple elementary proofs for some well-known Hankel determinants and their q-analogues.

Let $\{E_n\}$ be the Euler numbers. In the paper we determine $E_{2^mk+b}-E_b$ modulo $2^{m+7}$, where $k$ and $m$ are positive integers and $b\in{0,2,4,...}$.

The family of Shallit sequences consists of the Lucas sequences satisfying the recurrence $U_{n+2}(k)=(4k+2)U_{n+1}(k) -U_n(k),$ with initial values $U_0(k)=0$ and $U_1(k)=1$ and with $k\ge 1$ arbitrary. For every fixed $k$ the integers…

Hankel tensors arise from applications such as signal processing. In this paper, we make an initial study on Hankel tensors. For each Hankel tensor, we associate it with a Hankel matrix and a higher order two-dimensional symmetric tensor,…

In this paper, we use the tools of Gr\"{o}bner bases and combinatorial secant varieties to study the determinantal ideals $I_t$ of the extended Hankel matrices. Denote by $c$-chain a sequence $a_1,\...,a_k$ with $a_i+c<a_{i+1}$ for all…