Related papers: Hankel determinants and shifted periodic continued…
We evaluate the Hankel determinants of the convolution powers of Motzkin numbers for $r\leq 27$ by finding shifted periodic continued fractions, which arose in application of Sulanke and Xin's continued fraction method. We also conjecture…
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…
The Hankel determinants of a given power series $f$ can be evaluated by using the Jacobi continued fraction expansion of $f$. However the existence of the Jacobi continued fraction needs that all Hankel determinants of $f$ are nonzero. We…
For a single value of $\ell$, let $f(n,\ell)$ denote the number of lattice paths that use the steps $(1,1)$, $(1,-1)$, and $(\ell,0)$, that run from $(0,0)$ to $(n,0)$, and that never run below the horizontal axis. Equivalently, $f(n,\ell)$…
The Hankel determinants $\left(\frac{r}{2(i+j)+r}\binom{2(i+j)+r}{i+j}\right)_{0\leq i,j \leq n-1}$ of the convolution powers of Catalan numbers were considered by Cigler and by Cigler and Krattenthaler. We evaluate these determinants for…
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…
For any integer $m\geq 2$ and $r \in \{1,\dots, m\}$, let $f_n^{m,r}$ denote the number of $n$-Dyck paths whose peak's heights are $im+r$ for some integer $i$. We find the generating function of $f_n^{m,r}$ satisfies a simple algebraic…
We prove and generalize a conjecture of Johann Cigler on the Hankel determinants of convolution powers of Narayana polynomials. Our method follows a "guess-and-prove" strategy, relying on established techniques involving Hankel continued…
Cigler considered certain shifted Hankel determinants of convolution powers of Catalan numbers and conjectured identities for these determinants. Recently, Fulmek gave a bijective proof of Cigler's conjecture. Cigler then provided a…
In this expository paper we compute Hankel determinants of some sequences whose generating functions are given by C-fractions and derive orthogonality properties for associated polynomials.
In recent preprints, Cigler considered certain Hankel determinants of convoluted Catalan numbers and conjectured identities for these determinants. In this note, we shall give a bijective proof of Cigler's Conjecture by interpreting…
We evaluate Hankel determinants of matrices in which the entries are generating functions for paths consisting of up-steps, down-steps and level steps with a fixed starting point but variable end point. By specialisation, these determinant…
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 Euler numbers occur in the Taylor expansion of $\tan(x)+\sec(x)$. Since Stieltjes, continued fractions and Hankel determinants of the even Euler numbers, on the one hand, of the odd Euler numbers, on the other hand, have been widely…
This note presents some results about Hankel determinants of backwards shifted Catalan-like numbers and related sequences.
The $q$-analogs of Bernoulli and Euler numbers were introduced by Carlitz. Similar to the recent results on the Hankel determinants for the $q$-Bernoulli numbers established by Chapoton and Zeng, we determine parallel evaluations for the…
The middle binomial coefficients can be interpreted as numbers of Motzkin paths which have no horizontal steps at positive heights. Assigning suitable weights gives some nice polynomial extensions. We determine the Hankel determinants and…
Martin Aigner introduced Catalan-like numbers as elements of the first column of admissible matrices and studied Hankel determinants of their forward shifts. In this paper we collect some properties of the Hankel determinants of the other…
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…
Applying Johann Cigler's Hankel determinant formula in terms of the binomial coefficient determinants, which is simplified from Christian Krattenthale's, we get an explicit formula of Hankel determinants for general. As far as I know, those…