Related papers: On the Orthogonality of Generalized Pattern Sequen…
In contrast to finite arithmetic configurations, relatively little is known about which infinite patterns can be found in every set of natural numbers with positive density. Building on recent advances showing infinite sumsets can be found,…
For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to…
Partially ordered sets have received much attention in recent years, not just due to their usefulness in combinatorics and abstract algebra, but also due to their practical applications in fields ranging from chemistry to macroeconomics.…
We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…
In 2015, Phulara established a generalization of the famous central set theorem by an original idea. Roughly speaking, this idea extends a combinatorial result from one large subset of the given semigroup to countably many. In this paper,…
We consider series of the form $$ \frac{p}{q} +\sum_{j=2}^\infty \frac{1}{x_j}, $$ where $x_1=q$ and the integer sequence $(x_n)$ satisfies a certain non-autonomous recurrence of second order, which entails that $x_n|x_{n+1}$ for $n\geq 1$.…
A broader definition of generalized truncations of graphs is introduced followed by an exploration of some standard concepts and parameters with regard to generalized truncations.
The recent discoveries of new forms of quantum statistics require a close look at the under-lying Fock space structure. This exercise becomes all the more important in order to provide a general classification scheme for various forms of…
Patterns on numerical semigroups are multivariate linear polynomials, and they are said to be admissible if there exists a numerical semigroup such that evaluated at any nonincreasing sequence of elements of the semigroup gives integers…
We investigate the average number of representations of a positive integer as the sum of $k + 1$ perfect $k$-th powers of primes. We extend recent results of Languasco and the last Author, which dealt with the case $k = 2$ [6] and $k = 3$…
In the present work the existence of some patterns of primes is shown which generalize the celebrated result of Green and Tao according to which there are arbitrarily long arithmetic progressions in the sequence of primes
The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when…
We survey structures endowed with natural partial orderings and prove their universality. These partial orders include partial orders on sets of words, partial orders formed by geometric objects, grammars, polynomials and homomorphism order…
The pentagonal number theorem is extended to the sequence of the number of integer partitions with all parts equal. The new pentagonal number theorem implies that the distribution of the primes is just a specific detail of the application…
In this article we propose a general method of obtaining infinite sums of products with functions that count patterns in numbers.
A collection of integer sequences is jointly ergodic if for every ergodic measure preserving system the multiple ergodic averages, with iterates given by this collection of sequences, converge in the mean to the product of the integrals. We…
These lecture notes provide an informal introduction to the theory of nonnegative polynomials and sums of squares. We highlight the history and some recent developments, especially the new connections with classical (complex) algebraic…
This is the first in a set of three papers providing an introduction to generalised Cesaro convergence. We start with traditional Cesaro methods for extending classical convergence and further generalise these to allow the calculation of…
This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and…
The first author introduced a sequence of polynomials (\cite{8}, sequence A174531) defined recursively. One of the main results of this study is proof of the integrality of its coefficients.