Related papers: On properties of the generalized Davenport Expansi…

We prove a new generalization of Davenport's Fourier expansion of the infinite series involving the fractional part function over arithmetic functions. A new Mellin transform related to the Riemann zeta function is also established.

We introduced a new continued fraction expansions in our previous paper. For these expansions, we show formulae of probability about incomplete quotients. Furthermore, we prove the existence of invariant measures with respect to the…

The generalized divided differences are introduced. They are applied to investigate some properties characterizing generalized higher-order convexity. Among others some support-type property is proved.

We establish an explicit connection between a Davenport expansion and the Popov sum. Asymptotic analysis follows as a result of these formulas. New solutions to a query of N.J. Fine are offered, and a proof of Davenport expansions is…

We prove characterizations of Appell polynomials by means of symmetric property. For these polynomials, we establish a simple linear expression in terms of Bernoulli and Euler polynomials. As applications, we give interesting examples. In…

Many generalizations of continued fractions, where the reciprocal function has been replaced by a more general function, have been studied, and it is often asked whether such generalized expansions can have nice properties.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…

We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible structure of a matrix corresponding to the repetend, and use it…

Starting from the equation obeyed by the derivative, we construct several expansions of the solutions of the general Heun equation in terms of the Appell generalized hypergeometric functions of two variables of the fist kind. Several cases…

In this paper we study Appell polynomials by connecting them to random variables. This probabilistic approach yields, e.g., the mean value property which is fundamental in the sense that many other properties can be derived from it. We also…

The paper presents the generalized persistency of excitation conditions. They are not only valid for much broader range of applications than their classical counterparts but also elegantly prove the validity of the latter. The novelty and…

We prove a probabilistic Fourier extension theorem that says Fourier extension holds when averaged over certain smooth Alpert multipliers. The proofs use smooth Alpert wavelets with the classical techniques of stationary phase and…

We establish exponential inequalities for a class of V-statistics under strong mixing conditions. Our theory is developed via a novel kernel expansion based on random Fourier features and the use of a probabilistic method. This type of…

In this paper, we present the general one-dimensional Clifford Fourier Transform. We derive fundamental properties: Plancherel theorem, reconstruction and convolution formulas. Additionally, we provide an application to probability theory…

We use sequences which depend on two parameters to define families of ultradifferentiable functions which contain Gevrey classes. It is shown that such families are closed under superposition, and therefore inverse closed as well.…

We generalize Jacod's condition and introduce a new type sufficient condition for the uniform integrability of the general stochastic exponential.

In this paper, we introduce the polynomial continued fraction, a close relative of the well-known simple continued fraction expansions which are widely used in number theory and in general. While they may not possess all the intriguing…

Generalized Feller theory provides an important analog to Feller theory beyond locally compact state spaces. This is very useful for solutions of certain stochastic partial differential equations, Markovian lifts of fractional processes, or…

We formulate and prove the existence and uniqueness of the generalized Fourier transform associated with the absolutely continuous part of an arbitrary selfadjoint operator on a separable Hilbert space. To this end we develop a novel method…

Applying the approach based on the equation for the derivative, we construct several expansions of the solutions of the general Heun equation in terms of the incomplete Beta functions. Several expansions in terms of the Appell generalized…

A multivariate extension of the Dickman distribution was recently introduced, but very few properties have been studied. We discuss several properties with an emphasis on simulation. Further, we introduce and study a multivariate extension…