Related papers: An interesting class of Hankel determinants

We give a new and simple proof of the Hankel inversion formula for the classical Hankel transform which holds for a complex order with real part greater than -1. Using the proof of this formula we obtain the full description of the Kirillov…

In the paper, we give the recurrent equations of the Hankel determinants of the Cantor sequence, and show that the Hankel determinants as a double sequence is 3-automatic. With the help of the Hankel determinants, we prove that the…

An infinite $\pm 1$-sequence is called {\it Apwenian} if its Hankel determinant of order $n$ divided by $2^{n-1}$ is an odd number for every positive integer $n$. In 1998, Allouche, Peyri\`ere, Wen and Wen discovered and proved that the…

Let $A$ be a set in a prime field $\mathbb{F}_p$. In this paper, we prove that $d\times d$ matrices with entries in $A$ determine almost $|A|^{3+\frac{1}{45}}$ distinct determinants and almost $|A|^{2-\frac{1}{6}}$ distinct permanents when…

In this paper, our primary goal is to calculate the Hankel determinants for a class of lattice paths, which are distinguished by the step set consisting of \(\{(1,0), (2,0), (k-1,1), (-1,1)\}\), where the parameter \(k\geq 4\). These paths…

Focus in this paper is on the Hankel determinant, $H_3(1)$, for the well-known classes of bounded-turning, starlike and convex functions in the open unit disk $E=\{z\in \mathbb{C}\colon|z|<1\}$. The results obtained complete the series of…

We mainly introduce two new kinds of numbers given by $$R_n=\sum_{k=0}^n\binom nk\binom{n+k}k\frac1{2k-1}\quad\ (n=0,1,2,...)$$ and $$S_n=\sum_{k=0}^n\binom nk^2\binom{2k}k(2k+1)\quad\ (n=0,1,2,...).$$ We find that such numbers have many…

We study higher order determinantal varieties obtained by considering generic $m\times n$ ($m \le n$) matrices over rings of the form $F[t]/(t^k)$, and for some fixed $r$, setting the coefficients of powers of $t$ of all $r \times r$ minors…

We investigate the link between rectangular Jack polynomials and Hankel hyperdeterminants. As an application we give an expression of the even power of the Vandermonde in term of Jack polynomials.

In this paper we give improved, probably not sharp, upper bounds of the Hankel determinant of third order for various classes of univalent functions and conjecture the sharp one.

We continue with the study of the Hankel determinant, $$ D_{n}(t,\alpha):=\det\left(\int_{0}^{\infty}x^{j+k}w(x;t,\alpha)dx\right)_{j,k=0 }^{n-1}, $$ generated by singularly perturbed Laguerre weight, $$ w(x;t,\alpha):=x^{\alpha}{\rm…

Starting with a definition based on the Catalan numbers, we carry out an empirical study of the Rueppel sequence. We use the Hankel transform as the main technique. By means of this transform we find links to such sequences as the Jacobi…

We give a deterministic algorithm that, given a composite number $N$ and a target order $D \ge N^{1/6}$, runs in time $D^{1/2+o(1)}$ and finds either an element $a \in \mathbb{Z}_N^*$ of multiplicative order at least $D$, or a nontrivial…

Let $r\geq 1$ be an integer, $\mathbf a=(a_1,\ldots,a_r)$ a vector of positive integers and let $D\geq 1$ be a common multiple of $a_1,\ldots,a_r$. We study two natural determinants of order $rD$ with Bernoulli polynomials and we present…

This article evaluates the determinants of two classes of special matrices, which are both from a number theory problem. Applications of the evaluated determinants can be found in [arXiv:math.NT/0509523]. Note that the two determinants are…

In this paper we determine the asymptotics of the determinants of truncated Wiener-Hopf plus Hankel operators $\det(W_R(a)\pm H_R(a))$ as $R$ tends to infinity for symbols $a(x)=(x^2/(1+x^2))^\beta$ with the parameter $\beta$ being of small…

A homogeneous polynomial S(x_1, ..., x_n) of degree r in n variables posesses a discriminant D_{n|r}(S), which vanishes if and only if the system of equations dS/dx_i = 0 has non-trivial solutions. We give an explicit formula for…

Bernoulli numbers are usually expressed in terms of their lower index numbers (recursive). This paper gives explicit formulas for Bernoulli numbers of even index. The formulas contain a remarkable sequence of determinants. The value of…

We consider a particular type of matrices which belong at the same time to the class of Hessenberg and Toeplitz matrices, and whose determinants are equal to the number of a type of compositions of natural numbers. We prove a formula in…

Let $\mathcal{A}$ denote the class of analytic functions such that $f(0)=0$ and $f'(0)=1$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}$. In this paper, we discuss the properties of a starlike subclass and compute its second and…