Our goal is to study Pascal-Sierpinski gaskets, which are certain fractal sets defined in terms of divisibility of entries in Pascal's triangle. The principal tool is a carry rule for the addition of the base-q representation of coordinates of points in the unit square. In the case that q = p is prime, we connect the carry rule to the power of p appearing in the prime factorization of binomial coefficients. We use the carry rule to define a family of fractal subsets Bqr of the unit square, and we...

The classical system of functional equations      $1/n{\sum }_{\nu =0}^{n-1}F((x+\nu )/n)=n^{-s}F\left(x\right)$ (n ∈ ℕ) with s ∈ ℂ, investigated for instance by Artin (1931), Yoder (1975), Kubert (1979), and Milnor (1983), is extended to      $1/n{\sum }_{\nu =0}^{n-1}F((x+\nu )/n)={\sum }_{d=1}^{\infty }{\lambda }_n\left(d\right)F\left(dx\right)$ (n ∈ ℕ) with complex valued sequences ${\lambda }_n$. This leads to new results on the periodic integrable and the aperiodic continuous solutions F:ℝ₊ → ℂ interrelating the theory of functional equations and the theory of arithmetic functions.

Flajolet and Richmond have invented a method to solve a large class of divide-and-conquer recursions. The essential part of it is the asymptotic analysis of a certain generating function for $z\to \infty$ by means of the Mellin transform. In this paper this type of analysis is performed for a reasonably large class of generating functions fulfilling a functional equation with polynomial coefficients. As an application, the average life time of a party of $N$ people is computed, where each person advances one...

In this paper we investigate the asymptotic properties of all solutions of the delay differential equation y’(x)=a(x)y((x))+b(x)y(x),      xI=[x0,). We set up conditions under which every solution of this equation can be represented in terms of a solution of the differential equation z’(x)=b(x)z(x),      xI and a solution of the functional equation |a(x)|((x))=|b(x)|(x),      xI.

We discuss the asymptotic behaviour of all solutions of the functional differential equation $$y^{\text{'}}\left(x\right)=\sum _{i=1}^m{a}_i\left(x\right)y\left({\tau }_i\left(x\right)\right)+b\left(x\right)y\left(x\right),$$ where $b\left(x\right)<0$. The asymptotic bounds are given in terms of a solution of the functional nondifferential equation $$\sum _{i=1}^m\left|a_i\left(x\right)\right|\omega \left({\tau }_i\left(x\right)\right)+b\left(x\right)\omega \left(x\right)=0.$$

The paper discusses the asymptotic properties of solutions of the scalar functional differential equation $$y^{\text{'}}\left(x\right)=ay\left(\tau \left(x\right)\right)+by\left(x\right),x\in [x_0,\infty )$$ of the advanced type. We show that, given a specific asymptotic behaviour, there is a (unique) solution $y\left(x\right)$ which behaves in this way.

A lot is known about the forward iterates of an analytic function which is bounded by 1 in modulus on the unit disk D. The Denjoy-Wolff Theorem describes their convergence properties and several authors, from the 1880's to the 1980's, have provided conjugations which yield very precise descriptions of the dynamics. Backward-iteration sequences are of a different nature because a point could have infinitely many preimages as well as none. However, if we insist in choosing preimages that are at a...