Related papers: Hankel Determinant Solution for Elliptic Sequence
For small $r$ the Hankel determinants $d_r(n)$ of the sequence $\left({2n+r\choose n}\right)_{n\ge 0}$ are easy to guess and show an interesting modular pattern. For arbitrary $r$ and $n$ no closed formulae are known, but for each positive…
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…
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 compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discontinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support.…
We study the Hankel determinant generated by a Gaussian weight with Fisher-Hartwig singularities of root type at $t_j$, $j=1,\cdots ,N$. It characterizes a type of average characteristic polynomial of matrices from Gaussian unitary…
An isogeny class of elliptic curves over a finite field is determined by a quadratic Weil polynomial. Gekeler has given a product formula, in terms of congruence considerations involving that polynomial, for the size of such an isogeny…
An elliptic divisibility sequence is an integer recurrence sequence associated to an elliptic curve over the rationals together with a rational point on that curve. In this paper we present a higher-dimensional analogue over arbitrary base…
The aim of this paper is to obtain an upper bound to the second Hankel the determinant for starlike and convex functions of order.
An explicit characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy is presented. Our approach is based on (an extension of) a classical theorem of Picard, which guarantees the existence of solutions which are…
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…
In this note, we explore certain determinantal descriptions of the Robbins numbers. Techniques used for this include continued fractions, Riordan arrays and series inversion. Proven and conjectured representations involve the determinants…
One deals with degenerations by coordinate sections of the square generic Hankel matrix over a field $k$ of characteristic zero, along with its main related structures, such as the determinant of the matrix, the ideal generated by its…
As argued by Hone in the paper [Commun. Pure Appl. Math., 74(11):2310--2347, 2021], a ``mismatch" problem remained unsolved while he was investigating continued fraction expansions and Hankel determinants from hyperelliptic curves. In this…
In this paper, we study the finiteness problem of torsion points on an elliptic curve whose coordinates satisfy some multiplicative dependence relations. In particular, we prove that on an elliptic curve defined over a number field there…
We prove formulas for power moments for point counts of elliptic curves over a finite field $k$ such that the groups of $k$-points of the curves contain a chosen subgroup. These formulas express the moments in terms of traces of Hecke…
Upon solving a finite discrete reduction of the difference Heun equation, we arrive at an elliptic generalization of the Racah polynomials. We exhibit the three-term recurrence relation and the orthogonality relations for these elliptic…
We study the Hankel determinant for the weight $x^{\alpha}{\rm exp}(-x-t_1/x-t_2/x^2), x\in[0,+\infty)$, with $\alpha>-1,~t_1\in\mathbb{R}\setminus\{0\}, ~t_2>0.$ Compared with the weight $x^{\alpha}{\rm e}^{-x-t_1/x}$ studied in prior work…
This note collects some facts and conjectures about the Hankel determinants and their generating functions of the columns of Hoggatt triangles which apparently are related to combinatorial objects such as Young tableaux and Narayana…
We propose a conjectural construction of global points on modular elliptic curves over arbitrary number fields, generalizing both the p-adic construction of Heegner points via Cerednik-Drinfeld uniformization and the definition of classical…
We establish a determinant formula for the bilinear form associated with the elliptic hypergeometric integrals of type $BC_n$ by studying the structure of $q$-difference equations to be satisfied by them. The determinant formula is proved…