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Arithmetic based fractals associated with Pascal's triangle.

T.W. Gamelin, Mamiron A. Mnatsakanian (2005)

Publicacions Matemàtiques

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...

Arithmetical aspects of certain functional equations

Lutz G. Lucht (1997)

Acta Arithmetica

The classical system of functional equations       1 / n ∑ ν = 0 n - 1 F ( ( x + ν ) / n ) = n - s F ( x ) (n ∈ ℕ) with s ∈ ℂ, investigated for instance by Artin (1931), Yoder (1975), Kubert (1979), and Milnor (1983), is extended to       1 / n ∑ ν = 0 n - 1 F ( ( x + ν ) / n ) = ∑ d = 1 ∞ λ n ( d ) F ( d x ) (n ∈ ℕ) with complex valued sequences λ 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.

Asymptotic analysis of a class of functional equations and applications

P. J. Grabner, H. Prodinger, R. F. Tichy (1993)

Journal de théorie des nombres de Bordeaux

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 → ∞ 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...

Asymptotic behavior of solutions of nonlinear difference equations

Janusz Migda (2004)

Mathematica Bohemica

The nonlinear difference equation x n + 1 - x n = a n ϕ n ( x σ ( n ) ) + b n , ( E ) where ( a n ) , ( b n ) are real sequences, ϕ n ℝ ⟶ ℝ , ( σ ( n ) ) is a sequence of integers and lim n ⟶ ∞ σ ( n ) = ∞ , is investigated. Sufficient conditions for the existence of solutions of this equation asymptotically equivalent to the solutions of the equation y n + 1 - y n = b n are given. Sufficient conditions under which for every real constant there exists a solution of equation () convergent to this constant are also obtained.

Currently displaying 301 – 320 of 355