Related papers: Renewal-type Limit Theorem for Continued Fractions…

We establish a law of the iterated logarithm (LIL) for the set of real numbers whose $n$-th partial quotient is bigger than $\alpha_n$, where $(\alpha_n)$ is a sequence such that $\sum 1/\alpha_n$ is finite. This set is shown to have…

In this paper we study the existence and continuation of solution to general fractional differential equation with Hilfer fractional derivative. First we establish new local existence theorems. Then we derive the continuation theorems. With…

Let $\zeta(s,C)$ be the partial zeta function attached to a ray class C of a real quadratic field. We study this zeta function at s=1 and s=0, combining some ideas and methods due to Zagier and Shintani. The main results are (1) a…

V.I. Arnold has experimentally established that the limit of the statistics of incomplete quotients of partial continued fractions of quadratic irrationalities coincides with the Gauss--Kuz'min statistics. Below we briefly prove this fact…

For $\alpha_0 = \left[a_0, a_1, \ldots\right]$ an infinite continued fraction and $\sigma$ a linear fractional transformation, we study the continued fraction expansion of $\sigma(\alpha_0)$ and its convergents. We provide the continued…

Let $F$ be a distribution function on the line in the domain of attraction of a stable law with exponent $\alpha\in(0,1/2]$. We establish the strong renewal theorem for a random walk $S_1,S_2,\ldots$ with step distribution $F$, by extending…

This paper presents a reformulation of the Leibniz product rule as a finite sum that expresses the fractional derivative of the product of two differentiable functions. This paper then proves the cases for when the product consists of an…

In this work, we prove the joint convergence in distribution of $q$ variables modulo one obtained as partial sums of a sequence of i.i.d. square integrable random variables multiplied by a common factor given by some function of an…

In some recent papers, the authors considered regular continued fractions of the form \[ [a_{0};\underbrace{a,...,a}_{m}, \underbrace{a^{2},...,a^{2}}_{m}, \underbrace{a^{3},...,a^{3}}_{m}, ... ], \] where $a_{0} \geq 0$, $a \geq 2$ and $m…

We prove singularity of some distributions of random continued fractions that correspond to iterated function systems with overlap and a parabolic point. These arose while studying the conductance of Galton-Watson trees.

In this paper, we improve some transcendence results for $p$--adic continued fractions. In particular, we prove that palindromic and quasi--periodic $p$--adic continued fractions converge either to transcendental numbers or quadratic…

We study the generalized continued fraction expansions of complex numbers in term of elements from Euclidean subrings, especially Gaussian or Eisenstein integers, in a general framework as pursued in [3] and [1]. We introduce a common…

The prime aim of the present paper is to continue developing the theory of tempered fractional integrals and derivatives of a function with respect to another function. This theory combines the tempered fractional calculus with the…

Let $K$ be a number field. We show that, up to allowing a finite set of denominators in the partial quotients, it is possible to define algorithms for $\mathfrak P$-adic continued fractions satisfying the finiteness property on $K$ for…

We prove there exists a density one subset $\dd \subset \N$ such that each $n \in \dd$ is the denominator of a finite continued fraction with partial quotients bounded by 5.

In this paper, we formulate a regular $q$-fractional Sturm--Liouville problem (qFSLP) which includes the left-sided Riemann--Liouville and the right-sided Caputo $q$-fractional derivatives of the same order $\alpha$, $\alpha\in (0,1)$. We…

Let (g,k) be a reductive symmetric superpair of even type, i.e. so that there exists an even Cartan subspace a in p. The restriction map S(p^*)^k->S(a^*)^W where W=W(g_0:a) is the Weyl group, is injective. We determine its image explicitly.…

Three versions of the Freiheitssatz are proved in the context of one-relator quotients of limit groups, where the latter are equipped with 1-acylindrical splittings over cyclic subgroups. These are natural extensions of previously published…

Let $\frak E$ denote be the ring of Eisenstein integers. Let $z\in \mathbb C$ and $p_n,q_n \in \frak E$ be such that $\{p_n/q_n\}$ is the sequence of convergents corresponding to the continued fraction expansion of $z$ with respect to the…

Several continued fraction expansions for $e$ have been produced by an automated conjecture generator (ACG) called \emph{The Ramanujan Machine}. Some of these were already known, some have recently been proved and some remain unproven.…