Related papers: On certain maps defined by infinite sums

An identity is proved connecting two finite sums of inverse tangents. This identity is discretized version of Jacobi's imaginary transformation for the modular angle from the theory of elliptic functions. Some other related identities are…

The article is devoted to the alternating Cantor series. It is proved that any real number belonging to $[a_0-1;a_0]$, where $a_0=\sum^{\infty} _{k=1} {\frac{d_{2k}-1}{d_1d_2...d_{2k}}} $, has no more than two representations by the series…

Let X = S \oplus G, where S is a countable abelian semigroup and G is a countably infinite abelian group such that {2g : g in G} is infinite. Let pi: X \to G be the projection map defined by pi(s,g) = g for all x =(s,g) in X. Let f:X \to…

In a recent paper the authors studied the denominators of polynomials that represent power sums by Bernoulli's formula. Here we extend our results to power sums of arithmetic progressions. In particular, we obtain a simple explicit…

The problem of representation of elements of weighted space of infinitely differentiable functions on real line by exponential series is considered.

Infinite series of Bessel function of the first kind, $\sum_\nu^{\pm\infty} J_{N\nu+p}(x)$, $\sum_\nu^{\pm\infty} (-1)^\nu J_{N\nu+p}(x)$, are summed in closed form. These expressions are evaluated by engineering a Dirac comb that selects…

Cantor's ordinal numbers, a powerful extension of the natural numbers, are a cornerstone of set theory. They can be used to reason about the termination of processes, prove the consistency of logical systems, and justify some of the core…

We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concern colorings of graphs and bounded graphs,…

The nonlinear concepts of mixed summable families and maps for the spaces that only non-void sets are developed. Several characterizations of the corresponding concepts are achieved and the proof for a general Pietsch Domination-type…

In the first section of this paper we prove a theorem for the number of columns of a rectangular area that are identical to the given one. In the next section we apply this theorem to derive several combinatorial identities by counting…

We consider an arbitrary representation of the additive group over a field of characteristic zero and give an explicit description of a finite separating set in the corresponding ring of invariants.

A new integral representation is derived using a definite integral given by Cauchy and used to evaluate a number of integrals containing the finite series of special functions.

This paper presents a new version of boundary on coarse spaces. The space of ends functor maps coarse metric spaces to uniform topological spaces and coarse maps to uniformly continuous maps.

We prove, over any base ring, that the infinity-category of strictly unital A-infinity-categories (and strictly unital functors) is equivalent to the infinity-category of unital A-infinity-categories (and unital functors). We also identify…

The union of a collection of $n$ sets is generally expressed in terms of a characteristic (indicator) function that contains $2^{n}-1$ terms. In this article, a much simpler expression is found that requires the evaluation of $n$ terms…

Classical hypergeometric functions are well-known to play an important role in arithmetic algebraic geometry. These functions offer solutions to ordinary differential equations, and special cases of such solutions are periods of…

Infinitary Combinatorics shows interesting contrasts, with many similarities but also several important differences with its finite analog. The purpose of this paper is to present some concrete examples, both of similarities and of radical…

We characterize the indecomposable injective objects in the category of finitely presented representations of an interval finite quiver.

We consider the notion of the matrix (tensor) distribution of a measurable function of several variables. On the one hand, it is an invariant of this function with respect to a certain group of transformations of variables; on the other…

We define the syntax and reduction relation of a recursively typed lambda calculus with a parallel case-function (a parallel conditional). The reduction is shown to be confluent. We interpret the recursive types as information systems in a…