Related papers: A curious class of Hankel determinants
Hankel determinantal rings, i.e., determinantal rings defined by minors of Hankel matrices of indeterminates, arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves; they may also be viewed as…
In this paper we study some determinants and permanents. In particular, we investigate the new type determinants $$\det[(i^2+cij+dj^2)^{p-2}]_{1\le i,j\le p-1}\ \text{and} \ \det[(i^2+cij+dj^2)^{p-2}]_{0\le i,j\le p-1}$$ modulo an odd prime…
Let $f$ be analytic in the unit disk $\mathbb D$ and normalized so that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper, we give upper bounds of the Hankel determinant of second order for the classes of starlike functions of order $\alpha$,…
For any integer $m\ge 2$ and a set $V\subset \{1,\dots,m\}$, let $(m,V)$ denote the union of congruence classes of the elements in $V$ modulo $m$. We study the Hankel determinants for the number of Dyck paths with peaks avoiding the heights…
We derive a general expression for the Hankel determinants of a Dirichlet series F(s) and derive the asymptotic behavior for the special case that F(s) is the Riemann zeta function. In this case the Hankel determinant is a discrete analogue…
Recently I. Mezo studied a simple but interesting generalization of the exponential polynomials. In this note I consider two q-analogues of these polynomials and compute their Hankel determinants.
The Hankel determinant $H_{2,1}(F_{f^{-1}}/2)$ of logarithmic coefficients is defined as: \begin{align*} H_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} \Gamma_1 & \Gamma_2 \Gamma_2 & \Gamma_3 \end{vmatrix}=\Gamma_1\Gamma_3-\Gamma^2_2, \end{align*}…
We prove that the inverse of the Hankel matrix of the reciprocals of the Catalan numbers has integer entries. We generalize the result to an infinite family of generalized Catalan numbers. The Hankel matrices that we consider are associated…
Sulanke and Xin developed a continued fraction method that applies to evaluate Hankel determinants corresponding to quadratic generating functions. We use their method to give short proofs of Cigler's Hankel determinant conjectures, which…
We consider a family of all analytic and univalent functions (i.e., one-to-one) in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper, we obtain the sharp bounds of the second…
The symmetrized Slater determinants of orthogonal polynomials with respect to a non-negative Borel measure are shown to be represented by constant multiple of Hankel determinants of two other families of polynomials, and they can also be…
In this note we show that various natural q-analogues of the Catalan numbers can be obtained in a uniform way. Furthermore we compute their Hankel determinants.
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 define two new families of polynomials that generalize permanents and prove upper and lower bounds on their determinantal complexities comparable to the known bounds for permanents. One of these families is obtained by replacing…
To a sequence (s_n)_{n\ge 0} of real numbers we associate the sequence of Hankel matrices \mathcal H_n=(s_{i+j}),0\le i,j \le n. We prove that if the corresponding sequence of Hankel determinants D_n=\det\mathcal H_n satisfy D_n>0 for n<n_0…
It is a standard result that the Hankel determinants for a sequence stay invariant after performing the binomial transform on this sequence. In this work, we extend the scenario to $q$-binomial transforms and study the behavior of the…
In this note, we present the determinant, the inverse and a lower bound for the smallest eigenvalue for some Hankel matrices
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…
We give conjectures on the form of families of integer sequences whose Hankel transforms are, respectively, $(\alpha, \beta)$ Somos $4$ sequences, $(\alpha, 0, \gamma)$ Somos $6$ sequences, and $(\alpha, \beta, \gamma, \delta)$ Somos $8$…
In this paper we continue to investigate a certain class of Hankel-like positive definite kernels using their associated orthogonal polynomials. The main result of this paper is about the structure of this kind of kernels.