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Journals and Conferences
We discuss the partial infinite sum ∑∞ k=n u −s k for some positive integer n, where uk satisfies a recurrence relation of order s, un = aun−1 + un−2 + · · ·+ un−s (n ≥ s), with initial values u0 ≥ 0, uk ∈ N (0 ≤ k ≤ s− 1), where a and s(≥ 2) are positive integers. If a = 1, s = 2, and u0 = 0, u1 = 1, then uk = Fk is the k-th Fibonacci number. Our results… (More)
The stability of the functional equation F (x + y) − G(x − y) = 2H(x)K(y) over the domain of an abelian group G and the range of the complex field is investigated. Several related results extending a number of previously knownones, such as the ones dealingwith the sine functional equation, the d’Alembert functional equation and Wilson functional equation,… (More)
Rational arithmetic functions are arithmetic functions of the form g1 ∗···∗ gr ∗ h−1 1 ∗ ···∗h−1 s , where gi, hj are completely multiplicative functions and ∗ denotes the Dirichlet convolution. Four aspects of these functions are studied. First, some characterizations of such functions are established; second, possible Busche-Ramanujan-type identities are… (More)
where α∈R, and n=∏p primepp denotes the prime factorization of n. This function is called the generalized Möbius function because μ1 = μ, the wellknownMöbius function. Note that μ0 = I, the identity functionwith respect to Dirichlet convolution, μ−1 = ζ, the arithmetic zeta function and μα+β = μα∗μβ; α, β being real numbers. Recall that an arithmetic… (More)
In general, it is difficult to compute explicit solutions for nonlinear differential equations. In this note, we show how power function solutions can be computed for a class of nonlinear functional equations involving derivatives and iterates of the unknown functions.
Given two multiplicative arithmetic functions, various conditions for their convolution, powers , and logarithms to be completely multiplicative, based on values at the primes, are derived together with their applications.
A necessary and sufficient condition for two arithmetic functions to be linearly dependent over the set of prime-free functions is derived. A new kind of convolution is introduced and an application is given.
A generalized Ramanujan sum (GRS) is defined by replacing the usual Möbius function in the classical Ramanujan sum with the Souriau-Hsu-Möbius function. After collecting basic properties of a GRS, mostly containing existing ones, seven aspects of a GRS are studied. The first shows that the unique representation of even functions with respect to GRSs is… (More)