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Register a new user$q$-deformed rationals and $q$-continued fractions
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We introduce a notion of $q$-deformed rational numbers and $q$-deformed continued fractions. A $q$-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$-deformed Pascal identitiy for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the $q$-rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, $q$-deformation of the Farey graph, matrix presentations and $q$-continuants are given, as well as a relation to the Jones polynomial of rational knots.
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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 studied separately. However, no Hankel determinants of the (mixed) Euler numbers have been obtained and explicitly calculated. The reason for that is that some Hankel determinants of the Euler numbers are null. This implies that the Jacobi continued fraction of the Euler numbers does not exist. In the present paper, this obstacle is bypassed by using the Hankel continued fraction, instead of the $J$-fraction. Consequently, an explicit formula for the Hankel determinants of the Euler numbers is being derived, as well as a full list of Hankel continued fractions and Hankel determinants involving Euler numbers. Finally, a new $q$-analog of the Euler numbers $E_n(q)$ based on our continued fraction is proposed. We obtain an explicit formula for $E_n(-1)$ and prove a conjecture by R. J. Mathar on these numbers.
We reformulate several known results about continued fractions in combinatorial terms. Among them the theorem of Conway and Coxeter and that of Series, both relating continued fractions and triangulations. More general polygon dissections appear when extending these theorems for elements of the modular group $PSL(2,mathbb{Z})$. These polygon dissections are interpreted as walks in the Farey tessellation. The combinatorial model of continued fractions can be further developed to obtain a canonical presentation of elements of $PSL(2,mathbb{Z})$.
This is a translation of Eulers Latin paper De fractionibus continuis observationes into English. In this paper Euler describes his theory of continued fractions. He teaches, how to transform series into continued fractions, solves the Riccati-Differential equation by means of continued fractions and finds many other interesting formulas and results (e.g, the continued fraction for the quotient of two hypergeometric series usually attributed to Gau{ss})
We present an unexpected connection between two map enumeration problems. The first one consists in counting planar maps with a boundary of prescribed length. The second one consists in counting planar maps with two points at a prescribed distance. We show that, in the general class of maps with controlled face degrees, the solution for both problems is actually encoded into the same quantity, respectively via its power series expansion and its continued fraction expansion. We then use known techniques for tackling the first problem in order to solve the second. This novel viewpoint provides a constructive approach for computing the so-called distance-dependent two-point function of general planar maps. We prove and extend some previously predicted exact formulas, which we identify in terms of particular Schur functions.
If $q = p^n$ is a prime power, then a $d$-dimensional emph{$q$-Butson Hadamard matrix} $H$ is a $dtimes d$ matrix with all entries $q$th roots of unity such that $HH^* = dI_d$. We use algebraic number theory to prove a strong constraint on the dimension of a circulant $q$-Butson Hadamard matrix when $d = p^m$ and then explicitly construct a family of examples in all possible dimensions. These results relate to the long-standing circulant Hadamard matrix conjecture in combinatorics.
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