Related papers: On a Conjecture about Sums Involving Farey Fractio…
Inspired by a recently formulated conjecture by Bannai et al. we investigate spherical codes which admit exactly three different distances and are spherical 5-designs. Computing and analyzing distance distributions we provide new proof of…
Recently, Ni and Pan proved a $q$-congruence on certain sums involving central $q$-binomial coefficients, which was conjectured by Guo. In this paper, we give a generalization of this $q$-congruence and confirm another $q$-congruence, also…
Foulkes' conjecture has several generalisations due to Doran, Abdesselam--Chipalkatti, Bergeron, and Troyka. For the special linear Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, these assert that given $a \le c \le d \le b$ with $ab=cd$, the…
This article is part of an ongoing investigation of the two-dimensional Jacobian conjecture. In the first paper of this series, we proved the generalized Magnus' formula. In this paper, inspired by cluster algebras, we introduce a sequence…
We establish monotone bijections between the Farey sequences of order m and the halfsequences of Farey subsequences associated with the rank m elements of the Boolean lattice of subsets of a 2m-set. We also present a few related…
In this paper we construct the $q$-analogue of Barnes's Bernoulli numbers and polynomials of degree 2, for positive even integers, which is an answer to a part of Schlosser's question. For positive odd integers, Schlosser's question is…
We establish a congruence on sums of central $q$-binomial coefficients. From this $q$-congruence, we derive the divisibility of the $q$-trinomial coefficients introduced by Andrews and Baxter.
Kontsevich conjectured that the number f(G,q) of zeros over the finite field with q elements of a certain polynomial connected with the spanning trees of a graph G is polynomial function of q. We have been unable to settle Kontsevich's…
Ozeki and Prodinger showed that the odd power sum of the first several consecutive Fibonacci numbers of even order is equal to a polynomial evaluated at certain Fibonacci number of odd order. We prove that this polynomial and its derivative…
We disprove a 2002 conjecture of Dombi from additive number theory. More precisely, we find examples of sets $A \subset \mathbb{N}$ with the property that $\mathbb{N} \setminus A$ is infinite, but the sequence $n \rightarrow |\{ (a,b,c) \,…
In the first part of this paper we prove a conjecture of Hikami on the values of the radial limits of a family of $q$-hypergeometric false theta functions. Hikami conjectured that the radial limits are obtained by evaluating a truncated…
We outline an approach to prove the two dimensional Jacobian Conjecture using the theory of fractals.
We show that certain terminating $_{6}\phi_5$ series can be factorized into a product of two $_{3}\phi_{2}$ series. As applications we prove a summation formula for a product of two $q$-Delannoy numbers along with some congruences for sums…
By using the Rodriguez-Villegas-Mortenson supercongruences, we prove four supercongruences on sums involving binomial coefficients, which were originally conjectured by Sun. We also confirm a related conjecture of Guo on integer-valued…
In this PhD thesis we are dealing with convergence and summability of partial sums, Fej\'er and Marcinkiewicz means with respect to one- and two-dimensional Walsh-Fourier series on the martingale Hardy spaces. This thesis is focus to…
In this paper, we mainly prove the following conjectures of Z.-W. Sun \cite{S13}: Let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\in\mathbb{Z}$ and $x\equiv1\pmod 3$, then $$x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k}…
Seymour's distance two conjecture states that in any digraph there exists a vertex (a "Seymour vertex") that has at least as many neighbors at distance two as it does at distance one. We explore the validity of probabilistic statements…
Resolving a conjecture of Zhi-Wei Sun, we prove that every rational number can be represented as a sum of distinct unit fractions whose denominators are practical numbers. The same method applies to allowed denominators that are closed…
We prove that for all $q>61$, every non-zero element in the finite field $\mathbb{F}_{q}$ can be written as a linear combination of two primitive roots of $\mathbb{F}_{q}$. This resolves a conjecture posed by Cohen and Mullen.
The field diffracted from a one-dimensional, coherently illuminated periodic structure at fractional Talbot distances can be described as a coherent sum of shifted units cells weighted by a set of phases given by quadratic Gauss sums. We…