Related papers: Patterns in numbers and infinite sums and products
Finite trigonometric sums appear in various branches of Physics, Mathematics and their applications. For p; q to coprime positive integers and r we consider the finite trigonometric sums involving the product of three trigonometric…
In this short paper we present an elementary proof of the infinitude of primes. Our proof is similar in spirit to Euler's proof that the reciprocals of primes diverges and only uses tools from elementary number theory and calculus. In…
We use an elementary argument to prove some finite sums involving expressions of the forms $(q)_n$ and $(a;q)_n$ along with inductive formulas for some sequences.
An infinite permutation is a linear ordering of the set of non-negative integers. Generally, the properties of infinite permutations analogous to those of infinite words show some resemblances and some differences between permutations and…
The method of obtaining the set of noncanonical hypercomplex number systems by conversion of infinite hypercomplex number system to finite hypercomplex number system depending on multiplication rules and factorization method is described.…
One-parameter generalizations of the logarithmic and exponential functions have been obtained as well as algebraic operators to retrieve extensivity. Analytical expressions for the successive applications of the sum or product operators on…
We consider sequences of integers defined by a system of linear inequalities with integer coefficients. We show that when the constraints are strong enough to guarantee that all the entries are nonnegative, the generating function for the…
In this paper, a new method for generation of infinite series of symmetric identities written for exponential sums in real numbers is proposed. Such systems have numerous applications in theory of numbers, chaos theory, algorithmic…
Summation by parts is used to find the sum of a finite series of generalized harmonic numbers involving a specific polynomial or rational function. The Euler-Maclaurin formula for sums of powers is used to find the sums of some finite…
In this paper, we study the closed points of arithmetic schemes. We accomplish this by showing that the product of the cardinals of residue fields of closed points in an arithmetic scheme can be regularized. This regularization yields a new…
The article introduces the concept of uniformity, which is formulated as a scheme of axioms. The connection of this concept with ordered sets is studied. The effectiveness of using axiom schemes as a convenient and short way of replacing…
In this paper, we give explicit evaluation for some infinite series involving generalized (alternating) harmonic numbers. In addition, some formulas for generalized (alternating) harmonic numbers will also be derived.
Many identities written by $P=S=C$ are obtained, where $P$ infinite products, $S$ infinite series, and $C$ continued fractions. Such equality is called {\it triplicity}, and it can be used to compute the values of infinite series. It is…
It is known that the sum of the reciprocal of integers, $\sum_n (1/n)$, and the sum of the reciprocal of primes, $\sum_n (1/p_n)$, both diverge. Here, we study a series made from primes that sums exactly to 1. We also show this sum is…
In the present paper we propose generalizations of the regularity and counting lemmas for multidimensional matrices under a finite alphabet. Firstly, we prove a variant of a multidimensional regularity lemma with the help of a translation…
We present a new topological proof of the infinitude of prime numbers with a new topology. Furthermore, in this topology, we characterize the infinitude of any non-empty subset of prime numbers.
Ultraproducts are a well-known tool in the classical model theory of first-order logic. We explore their uses in the context of finite model theory.
We prove that the greedy sum of a direct product of two numeric arrays of complex numbers is equal to the product of the greedy sums of the factors provided that all the mentioned sums exist.
This paper deals with a new kind of generalized functions, called "ultrafunctions" which have been introduced recently and developed in some previous works. Their peculiarity is that they are based on a Non-Archimedean field namely on a…
Patterns describe proven solutions for recurring problems. Typically, patterns in a particular domain are interrelated and organized in pattern languages. As real-world problems often require patterns of multiple domains, different pattern…