Related papers: Generalized Pell's equations and Weber's class num…

In this paper, we study a class of sequences of polynomials linked to the sequence of Bell polynomials. Some sequences of this class have applications on the theory of hyperbolic differential equations and other sequences generalize…

Many results that are difficult can be found more easily by using a generalization in the complex plane of Einstein's addition law of parallel velocities. Such a generalization is a natural way to add quantities that are limited to bounded…

Based on the concepts of a generalized critical point and the corresponding generalized P.S. condition introduced by Duong Minh Duc[1], we have proved a new $Z_2$ index theorem and get a result on multiplicity of generalized critical…

A construction of new sequences of generalized Bernoulli polynomials of first and second kind is proposed. These sequences share with the classical Bernoulli polynomials many algebraic and number--theoretical properties. A new class of…

We give one parameter generalizations of the Fibonacci and Lucas numbers denoted by $\{F_n(\th)\}$ and $\{L_n(\th)\}$, respectively. We evaluate the Hankel determinants with entries $\{1/F_{j+k+1}(\th): 0\le i,j \le n\}$ and…

In this paper we introduce the generalization of Multi Poly-Euler polynomials and we investigate some relationship involving Multi Poly-Euler polynomials. Obtaining a closed formula for generalization of Multi Poly-Euler numbers therefore…

For each integer $m\ge3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. Given positive integers $a,b,c,k$ and an odd prime number $p$ with $p\nmid c$, we employ the theory of ternary…

Given a polynomial $f(x_1,x_2,\ldots, x_t)$ in $t$ variables with integer coefficients and a positive integer $n$, let $\alpha(n)$ be the number of integers $0\leq a<n$ such that the polynomial congruence $f(x_1, x_2, \ldots, x_t)\equiv a\…

The symmetry group method is applied to a generalized Korteweg-de Vries equation and several classes of group invarint solution for it are obtained by means of this technique. Polynomial, trigonometric and elliptic function solutions can be…

We present a method for the solution of polynomial equations. We do not intend to present one more method among several others, because today there are many excellent methods. Our main aim is educational. Here we attempt to present a method…

In this paper we produce a few continuations of our previous work on partitions into fractions. Specifically, we study strictly increasing integer sequences $\{n_j\}$ such that there are partitions for all integers less than the floor of…

Let $\alpha,\beta,\gamma\in\mathbb{N}$. We prove that given an $r$-colouring of $\mathbb{F}_p$ with $p$ prime, there are more than $c_{r,\alpha,\beta,\gamma} p^2$ solutions to the equation $x^\alpha+y^\beta=z^\gamma$ with all of $x,y,z$ of…

We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of…

The main purpose of this paper is solve polynomial equations that are satisfied by (generalized) polynomials. More exactly, we deal with the following problem: let $\mathbb{F}$ be a field with $\mathrm{char}(\mathbb{F})=0$ and $P\in…

We construct a random model to study the distribution of class numbers in special families of real quadratic fields $\mathbb Q(\sqrt d)$ arising from continued fractions. These families are obtained by considering periodic continued…

A generalization of the generating function for Gegenbauer polynomials is introduced whose coefficients are given in terms of associated Legendre functions of the second kind. We discuss how our expansion represents a generalization of…

Let $n \in \mathbb{Z}_{\geq 2}$. We study the generalized Fermat equation \[x^{13}+y^{13}=z^n, \quad x,y,z \in \mathbb{Z}, \quad \gcd(x,y,z)=1.\] Using a combination of techniques, including the modular method, classical descent, unit…

We present a multi-parameter non-constant-invariant class of Abel ordinary differential equations with the following remarkable features. This one class is shown to unify, that is, contain as particular cases, all the integrable classes…

In the present paper we generalize the Eulerian numbers (also of the second and third orders). The generalization is connected with an autonomous first-order differential equation, solutions of which are used to obtain integral…

In this paper, we establish an identity for Bernoulli's generalized polynomials. We deduce generalizations for many relations involving classical Bernoulli numbers or polynomials. In particular, we generalize a recent Gessel identity.