Related papers: Patterns in numbers and infinite sums and products
A decomposition of a natural number n is a sequence of consecutive natural numbers that sums to n. We construct a one-to-one correspondence between the odd factors of a natural number and its decompositions. We study the decompositions by…
Here we generalize the concept of spatial tensor product, introduced by Skeide, of two product systems via a pair of normalized units. This new notion is called amalgamated tensor product of product systems, and now the amalgamation can be…
Combining the derivative operator with a binomial sum from the telescoping method, we establish a family of summation formulas involving generalized harmonic numbers.
A recursive algorithm is constructed which finds all solutions to a class of Diophantine equations connected to the problem of determining ordered n-tuples of positive integers satisfying the property that their sum is equal to their…
In this article a new method of generating sums of like powers is presented.
The need to describe abrupt changes or response of nonlinear systems to impulsive stimuli is ubiquitous in applications. Also the informal use of infinitesimal and infinite quantities is still a method used to construct idealized but…
We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage…
The paper considers a universal approach that allows one to quite simply obtain nonlinear asymptotic estimates of various summation functions. It is shown the application of this approach to the asymptotic estimation of divergent Dirichlet…
In this article we introduce a new approach to compute infinite products defined by automatic sequences involving the Thue-Morse sequence. As examples, for any positive integers $q$ and $r$ such that $0 \leq r \leq q-1$, we find infinitely…
Using an application of Schmidt's Subspace Theorem, this paper gives new transcendence criteria for rapidly converging infinite products of algebraic numbers. The paper also improves existing criteria for irrationality of products and…
Two results on product of compact filters are shown to be the common principle behind a surprisingly large number of theorems.
Many mathematical statements have the following form. If something is true for all finite subsets of an infinite set $I$, then it is true for all of $I$. This paper describes some old and new results on infinite sets of linear and…
Short nonstandard proofs are given for some results about infinite systems of equations in infinitely many variables.
Generalized product formulas and index transforms, involving products of Whittaker's functions of different indices are established and investigated. The corresponding inversion formulas are found. Particular cases cover index transforms…
Given finitely many consecutive terms of an infinite sequence, we discuss the construction of a polynomial difference equation that the sequence may satisfy. We also present a method to seek a candidate polynomial differential equation for…
This paper introduces the target sum function along with its characteristics. The target sum function takes a list of integers and a specific target integer as input values and expresses the number of ways to obtain the target sum by either…
This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a…
In this short paper, we give a complete and affirmative answer to a conjecture on matrix trace inequalities for the sum of positive semidefinite matrices. We also apply the obtained inequality to derive a kind of generalized Golden-Thompson…
There is not much that can be said for all $x$ and for all $n$ about the sum \[ \sum_{k=1}^n \frac{1}{|\sin k\pi x|}. \] However, for this and similar sums, series, and products, we can establish results for almost all $x$ using the tools…
We study the concept of universal sets from the additive--combinatorial point of view. Among other results we obtain some applications of this type of uniformity to sets avoiding solutions to linear equations, and get an optimal upper bound…