Related papers: Hankel determinants of the Cantor sequence
The article is devoted to the investigation of representation of rational numbers by Cantor series. Necessary and sufficient conditions for a rational number to be representable by a positive Cantor series are formulated for the case of an…
In 1975 K. Michael Day produced an exact formula for the determinants of finite Toeplitz matrices whose symbols are rational. The answer is a sum that involves powers of the roots of the numerator of the symbol and whose coefficients depend…
The purpose of this paper is to describe asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices with singular generating functions. The formulas are similar to those of the analogous problem for finite Toeplitz…
Using a novel rewriting problem, we show that several natural decision problems about finite automata are undecidable (i.e., recursively unsolvable). In contrast, we also prove three related problems are decidable. We apply one result to…
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…
We further study the orthogonal polynomials with respect to the generalized Airy weight based on the work of Clarkson and Jordaan [{\em J. Phys. A: Math. Theor.} {\bf 54} ({2021}) {185202}]. We prove the ladder operator equations and…
The hyperdeteminants considered here are the simplest analogues of determinants for higher rank tensors which have been defined by Cayley, and apply only to tensors with an even number of indices. We have shown in a previous article that…
In the present article, modeling certain rational numbers, that are represented in terms of Cantor series, are described. The statements on relations between digits in the representations of rational numbers by Cantor series (for the case…
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…
We determine the sets definable in expansions of the ordered real additive group by generalized Cantor sets. Given a natural number $r\geq 3$, we say a set $C$ is a generalized Cantor set in base $r$ if there is a non-empty…
Let $\theta$ and $\rho$ be real numbers with $0 \le \theta, \rho < 1$ and $\theta$ irrational. We show that the Hecke-Mahler series $$ F_{\theta, \rho} (z_1, z_2) = \sum_{k_1 \ge 1} \, \sum_{k_2 = 1}^{\lfloor k_1 \theta + \rho \rfloor} \,…
In this paper, we obtain the upper bounds to the third Hankel determinants for starlike functions of order $\alpha$, convex functions of order $\alpha$ and bounded turning functions of order $\alpha$. Furthermore, several relevant results…
In this paper we will formulate $4\times4$ Riemann-Hilbert problems for Toeplitz+Hankel determinants and the associated system of orthogonal polynomials, when the Hankel symbol is supported on the unit circle and also when it is supported…
We give a brief history of transcendental number theory, including Schanuel's conjecture (S). Assuming (S), we prove that if z and w are complex numbers, not 0 or 1, with z^w and w^z algebraic, then z and w are either both rational or both…
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 give simple new proofs of some Hankel determinant evaluations by Omer Egecioglu and Aleksandar Cvetkovic, Predrag Rajkovic and Milos Ivkovic and prove analogous results for sums of moments of symmetric orthogonal polynomials.
Sequences of numbers (either natural integers, or integers or rational) of level $k \in \mathbb{N}$ have been defined in \cite{Fra05,Fra-Sen06} as the sequences which can be computed by deterministic pushdown automata of level $k$. This…
Nous \'etendons aux courbes de genre arbitraire le th\'eor\`eme de rationalit\'e de Cantor, lui-m\^eme une extension de th\'eor\`emes de Borel, P\'olya, Dwork, Bertrandias et Robinson. La d\'emonstration s'effectue en deux \'etapes. La…
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…
It is shown that every two-variable adjunction in categories enriched in a commutative quantale serves as a base for constructing Isbell adjunctions between functor categories, and Kan adjunctions are precisely Isbell adjunctions…