Related papers: Hankel determinants for convolution powers of Cata…
We discuss the properties of the Hankel transformation of a sequence whose elements are the sums of consecutive generalized Catalan numbers and find their values in the closed form.
The irrationality exponent of an irrational number $\xi$, which measures the approximation rate of $\xi$ by rationals, is in general extremely difficult to compute explicitly, unless we know the continued fraction expansion of $\xi$.…
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…
We study the Hankel determinants associated with the weight $$w(x;t)=(1-x^2)^{\beta}(t^2-x^2)^\alpha h(x),~x\in(-1,1),$$ where $\beta>-1$, $\alpha+\beta>-1$, $t>1$, $h(x)$ is analytic in a domain containing $[-1,1]$ and $h(x)>0$ for…
An extensive literature exists describing various techniques for the evaluation of Hankel determinants. The prevailing methods such as Dodgson condensation, continued fraction expansion, LU decomposition, all produce product formulas when…
For $ t \in [0,1]$ let $\underline{H}_{2\lfloor nt \rfloor} = ( m_{i+j})_{i,j=0}^{\lfloor nt \rfloor} $ denote the Hankel matrix of order $2\lfloor nt \rfloor$ of a random vector $(m_1,\ldots ,m_{2n})$ on the moment space…
We prove several evaluations of determinants of matrices, the entries of which are given by the recurrence $a_{i,j}=a_{i-1,j}+a_{i,j-1}$, or variations thereof. These evaluations were either conjectured or extend conjectures by Roland…
Hankel determinants and automatic sequences are two classical subjects widely studied in Mathematics and Theoretical Computer Science. However, these two topics were considered totally independently, until in 1998, when Allouche,…
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…
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 one parameter generalizations of the Fibonacci and Lucas numbers denoted by $\{F_n(\th)\}$ and $\{L_n(\th)\}$, respectively. We evaluate the Hankel determinants with entries $\{1/F_{j+k+1}(\th): 0\le i,j \le n\}$ and…
In this paper our purpose is to find upper bound estimate for the second Hankel determinant $|a_{2}a_{4}-a_{3}^{2}|$ for functions defined by convolution belonging to the class $\mathcal{N}_{\sigma}^{\mu,\delta}(\lambda,t)$ by using…
We give an overview of known results about Hilbert matrices from the point of view of orthogonal polynomials and compute Hankel determinants of harmonic numbers and related topics.
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…
We study the Hankel determinant generated by the moments of the deformed Laguerre weight function $x^{\alpha}{\rm{e}}^{-x}\prod\limits_{k=1}^{N}(x+t_k)^{\lambda_k}$, where $x\in \left[0,+\infty \right)$, $\alpha,t_k >0,…
Following van der Poorten, we consider a family of nonlinear maps which are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus $\mathrm{g}$. Using the connection with the classical theory of…
The aim of this paper is to obtain an upper bound to the second Hankel the determinant for starlike and convex functions of order.
In 1998, Allouche, Peyri\`ere, Wen and Wen considered the Thue--Morse sequence, and proved that all the Hankel determinants of the period-doubling sequence are odd integral numbers. We speak of $t$-extension when the entries along the…
The Hankel determinant $H_{2,1}(F_{f}/2)$ is defined as: \begin{align*} H_{2,1}(F_{f}/2):= \begin{vmatrix} \gamma_1 & \gamma_2 \gamma_2 & \gamma_3 \end{vmatrix}, \end{align*} where $\gamma_1, \gamma_2,$ and $\gamma_3$ are the first, second…
We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes iff n is composite. If the dimension is a prime p, then the…