Pseudo-factorials, elliptic functions, and continued fractions

This study presents miscellaneous properties of pseudo-factorials, which are numbers whose recurrence relation is a twisted form of that of usual factorials. These numbers are associated with special elliptic functions, most notably, a Dixonian and a Weierstrass function, which parametrize the Fermat cubic curve and are relative to a hexagonal lattice. A continued fraction expansion of the ordinary generating function of pseudo-factorials, first discovered empirically, is established here. This article also provides a characterization of the associated orthogonal polynomials, which appear to form a new family of "elliptic polynomials", as well as various other properties of pseudo-factorials, including a hexagonal lattice sum expression and elementary congruences.Comment: 24 pages; with correction of typos and minor revision. To appear in The Ramanujan Journa

Combinatorial properties of multidimensional continued fractions

The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization arising from an algorithm due to Jacobi, have been poorly investigated in this sense, up to now. In this paper, we propose a combinatorial interpretation of the convergents of multidimensional continued fractions in terms of counting some particular tilings, generalizing some results that hold for classical continued fractions

Proofs of two conjectures of Kenyon and Wilson on Dyck tilings

Recently, Kenyon and Wilson introduced a certain matrix $M$ in order to compute pairing probabilities of what they call the double-dimer model. They showed that the absolute value of each entry of the inverse matrix $M^{-1}$ is equal to the number of certain Dyck tilings of a skew shape. They conjectured two formulas on the sum of the absolute values of the entries in a row or a column of $M^{-1}$. In this paper we prove the two conjectures. As a consequence we obtain that the sum of the absolute values of all entries of $M^{-1}$ is equal to the number of complete matchings. We also find a bijection between Dyck tilings and complete matchings.Comment: 18 pages, 9 figure

Laurent Polynomials and Superintegrable Maps

This article is dedicated to the memory of Vadim Kuznetsov, and begins with some of the author's recollections of him. Thereafter, a brief review of Somos sequences is provided, with particular focus being made on the integrable structure of Somos-4 recurrences, and on the Laurent property. Subsequently a family of fourth-order recurrences that share the Laurent property are considered, which are equivalent to Poisson maps in four dimensions. Two of these maps turn out to be superintegrable, and their iteration furnishes infinitely many solutions of some associated quartic Diophantine equations

The Graham-Knuth-Patashnik recurrence: symmetries and continued fractions

Acompaña: Supplementary material for The Graham-Knuth-Patashnik recurrence: symmetries and continued fractionsThis research was supported in part by the Spanish MINECO grant FIS2014-57387-C3-3-P; by the FEDER/Ministerio de Ciencia, Innovación y Universidades-Agencia Estatal de Investigación grant FIS2017-84440-C2-2-P; by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23) and in the context of the V-PRICIT (Regional Plan for Scientific Research and Technological Innovation); and by U.K. Engineering and Physical Sciences Research Council grant EP/N025636/1.Publicad