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Consider the representation of a rational number as a continued fraction, associated with "odd" Euclidean algorithm. In this paper we prove certain properties for the limit distribution function for sequences of rationals with bounded sum…
We investigate a random integral which provides a natural example of an imaginary exponential functional of Brownian motion. This functional shows up in the study of the binary annihilation process, within the Doi-Peliti formalism for…
The principal aim of this article is to establish an iteration method on the space of resurgent functions. We discuss endless continuability of iterated convolution products of resurgent functions and derive their estimates developing the…
In this article, we present relations for the Euler totient function $\varphi(n)$ and the number of divisors $\tau(n)$ in terms of finite sums of integer parts of rational numbers or greatest common divisors of pairs of integers. Some of…
The density distribution function of the integral of the exponential Brownian motion is determined explicitly in the form of a rapidly convergent series.
We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional…
Many generalizations of continued fractions, where the reciprocal function has been replaced by a more general function, have been studied, and it is often asked whether such generalized expansions can have nice properties. For instance, we…
This paper sets the groundwork for the consideration of families of recursively defined polynomials and rational functions capable of describing the Bernoulli numbers. These families of functions arise from various recursive definitions of…
Let $(G_n)_{n\geqslant 0}$ be a linear recurrence sequence defining a numeration system and satisfying mild structural hypotheses. For real-valued G-additive functions (additive in the greedy G-digits), we establish an…
We define a triangular array closely related to Stern's diatomic array and show that for a fixed integer $r\geq 1$, the sum $u_r(n)$ of the $r$th powers of the entries in row $n$ satisfy a linear recurrence with constant coefficients. The…
The paper contains a combinatorial theorem (the sequence of Newton polygons of a reccurent sequence of polynomials is quasi-linear) and two applications of it in classical and quantum topology, namely in the behavior of the $A$-polynomial…
We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. The approximation of continuous bounded functions by entire functions of exponential type…
The restoration of an additive function defined on P parallelepipeds via its derivative with respect to P parallelepipeds is studied. The obtained theorem is applied to the questions of uniqueness of multiple series with regard to Haar and…
We provide examples of multiplicative functions $f$ supported on the squarefree integers, such that on primes $f(p)=\pm1$ and such that $M_f(x):=\sum_{n\leq x} f(n)=o(\sqrt{x})$. Further, by assuming the Riemann hypothesis (RH) we can go…
In this work at first the relation the Mittag-Lefler function to the exponential is given. The results are applied to the construction of the solution of Cauchy problem for ordinary linear operator differential equations with constant…
In this article, we present a binary tree with vertices given by rational functions $p(x)/q(x)$; the root and functional derivation of children are inspired by continued fractions. We prove some special properties of the tree. For example,…
We elaborate the notions of Martin-L\"of and Schnorr randomness for real numbers in terms of uniform distribution of sequences. We give a necessary condition for a real number to be Schnorr random expressed in terms of classical uniform…
We present an extension to the $\mathtt{mathlib}$ library of the Lean theorem prover formalizing the foundations of computability theory. We use primitive recursive functions and partial recursive functions as the main objects of study, and…
We prove the irrationality of some factorial series. To do so we combine methods from elementary and analytic number theory with methods from the theory of uniform distribution.
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…