Related papers: Catalan and Motzkin Integral Representations
We establish a connection between constructible representations (arising in the study of left cells in Weyl groups) and Catalan numbers.
We give formulas for the number of representations of non negative integers by various quadratic forms. We also give evaluations in the case of sum of two cubes (cubic case) and the quintic case, as well. We introduce a class of generalized…
An integer sequence that is defined by initial values and a linear recurrence with constant integer coefficients, can be represented by the difference of two arithmetic terms containing exponentiation. All constants occuring in the term are…
A proof for a conjecture by Shadrin and Zvonkine, relating the entries of a matrix arising in the study of Hurwitz numbers to a certain sequence of rational numbers, is given. The main tools used are iteration matrices of formal power…
An analytical-numeric calculation method of extremely complicated integrals is presented. These integrals appear often in magnet soliton theory. The appropriate analytical continuation and a corresponding integration contour allow to reduce…
In a recent beautiful but technical article, William Y.C. Chen, Qing-Hu Hou, and Doron Zeilberger developed an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequences,…
It is proved that all recursively enumerable sets of natural numbers can be represented by arithmetic formulas (of two kinds) with only 3 quantifiers.
By using Cauchy integral formula in the theory of complex functions, the authors establish some integral representations for the principal branches of several complex functions involving the logarithmic function, find some properties, such…
We construct (q,t)-Catalan polynomials and q-Fuss-Catalan polynomials for any irreducible complex reflection group W. The two main ingredients in this construction are Rouquier's formulation of shift functors for the rational Cherednik…
The Catalan triangle, as well as a Fuss-Catalan triangle, enter a problem of counting particular tied arc diagrams. This setting allows us to prove some combinatorial properties of these triangles.
Using modular forms we determine formulas for the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients $1$, $2$, $3$ or $6$.
Throughout history, recreational mathematics has always played a prominent role in advancing research. Following in this tradition, in this paper we extend some recent work with crazy sequential representations of numbers- equations made of…
In this (mostly expository) paper I want to share some observations prompted by a class of matrices whose determinants are Catalan numbers. Considering different methods of proof we obtain some generalizations and q-analogues and…
This paper explores generalized slice monogenic functions by introducing their operator symbols, representation formula, and integral formula. The study extends the Teodorescu transform to a broader class of theorems and inferences,…
We introduce a proper multi-type display calculus for bilattice logic (with conflation) for which we prove soundness, completeness, conservativity, standard subformula property and cut-elimination. Our proposal builds on the product…
This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials…
Recent developments in the categorical foundations of universal algebra have given fresh impetus to an understanding of the lambda calculus coming from categorical logic: an interpretation is a semi-closed algebraic theory. Scott's…
This paper focuses on a wide class of Collatz-type arithmetic dynamics, and presents a systematic derivation of recursive formulas and functional equations satisfied by the associated generating functions. The main tools belong to complex…
We show that the super Catalan numbers are special values of the Krawtchouk polynomials by deriving an expression for the super Catalan numbers in terms of a signed set.
In this paper we provide a complete approach to the real numbers via decimal representations. Construction of the real numbers by Dedekind cuts, Cauchy sequences of rational numbers, and the algebraic characterization of the real number…