On the Dynamic Geometry of Kasner Quadrilaterals with Complex Parameter

We explore the dynamics of the sequence of Kasner quadrilaterals (AnBnCnDn)n≥0 defined via a complex parameter α. We extend the results concerning Kasner triangles with a fixed complex parameter obtained in earlier works and determine the values of α for which the generated dynamics are convergent, divergent, periodic, or dense

An equivalent property of a Hilbert-type integral inequality and its applications

Making use of complex analytic techniques as well as methods involving weight functions, we study a few equivalent conditions of a Hilbert-type integral inequality with nonhomogeneous kernel and parameters. In the form of applications we deduce a few equivalent conditions of a Hilbert-type integral inequality with homogeneous kernel, and we additionally consider operator expressions

Remarks on the Coefficients of Inverse Cyclotomic Polynomials

Cyclotomic polynomials play an imporant role in discrete mathematics. Recently, inverse cyclotomic polynomials have been defined and investigated. In this paper, we present some recent advances related to the coefficients of inverse cyclotomic polynomials, including a practical recursive formula for their calculation and numerical simulations

On Cyclotomic Polynomial Coefficients

For a positive integer n > 1 the n-th cyclotomic polynomial is defined by (Formula presented) where ς are the primitive n-th roots of unity. These polynomials are known to possess many interesting properties. In this article we establish an integral formula for the coefficients of the cyclotomic polynomial, we then discuss the direct and alternate sums of coefficients, as well as the mid-term of Φn(z). Finally, these results are used in the computation of certain trigonometric integrals

On some new results for the generalised Lucas sequences

In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value x and, under certain conditions, we derive exact formulae (Theorems 3 and 4) and establish asymptotic limits for them (Theorem 6). We formulate necessary and sufficient arithmetic conditions which can identify the terms of a-Fibonacci and a-Lucas sequences. Finally, using a deep theorem of Siegel, we show that the aforementioned sequences contain only finitely many perfect powers. During the process we also discover some novel integer sequences

Weak Pseudoprimality Associated with the Generalized Lucas Sequences

Pseudoprimes are composite integers which share properties of the prime numbers, and they have applications in many areas, as, for example, in public-key cryptography. Numerous types of pseudoprimes are known to exist, many of them defined by linear recurrent sequences. In this material, we present some novel classes of pseudoprimes related to the generalized Lucas sequences. First, we present some arithmetic properties of the generalized Lucas and Pell–Lucas sequences and review some classical pseudoprimality notions defined for Fibonacci, Lucas, Pell, and Pell–Lucas sequences and their generalizations. Then we define new notions of pseudoprimality which do not involve the use of the Jacobi symbol and include many classical pseudoprimes. For these, we present associated integer sequences recently added to the Online Encyclopedia of Integer Sequences, identify some key properties, and propose a few conjectures

DYNAMIC GEOMETRY OF KASNER TRIANGLES WITH A FIXED WEIGHT

In this article we consider the geometrical iterative process defined by the Kasner triangles with a fixed real weight. The main results about the convergence of this process are given in Section 2. The proofs are elementary and use the general theory of the second order linear recurrences. In the remark after the proof a higher level approach is presented

On Two Kinds of the Hardy-Type Integral Inequalities in the Whole Plane with the Equivalent Forms

By the use of weight functions, a few equivalent conditions of two kinds of Hardy-type integral inequalities with multi-parameters in the whole plane are obtained. The constant factors related to the extended Riemann-zeta function are proved to be the best possible. Applying our results, we deduce a few equivalent conditions of two kinds of Hardy-type integral inequalities in the whole plane and some particular cases

On a Hilbert-type integral inequality in the whole plane with the equivalent forms

In the present paper we establish a few equivalent conditions of a Hilbert-type integral inequality with a non-homogeneous kernel in the whole plane. A few equivalent conditions of a Hilbert-type integral inequality with the homogeneous kernel in the whole plane are deduced, in the form of applications. We additionally consider operator expressions and several interesting particular cases

The number of partitions of a set and Superelliptic Diophantine equations

In this chapter we start by presenting some key results concerning the number of ordered k-partitions of multisets with equal sums. For these we give generating functions, recurrences and numerical examples. The coefficients arising from these formulae are then linked to certain elliptic and superelliptic Diophantine equations, which are investigated using some methods from Algebraic Geometry and Number Theory, as well as specialized software tools and algorithms. In this process we are able to solve some recent open problems concerning the number of solutions for certain Diophantine equations and to formulate new conjectures.N/