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Using the generalized Möbius functions, µ α , first introduced by Hsu (1995), two characterizations of completely multiplicative functions are given; save a minor condition they read (µ α f) −1 = µ −α f and f α = µ −α f .
We discuss the partial infinite sum ∞ k=n u −s k for some positive integer n, where u k satisfies a recurrence relation of order s, u n = au n−1 + u n−2 + · · · + u n−s (n ≥ s), with initial values u 0 ≥ 0, u k ∈ N (0 ≤ k ≤ s − 1), where a and s(≥ 2) are positive integers. If a = 1, s = 2, and u 0 = 0, u 1 = 1, then u k = F k is the k-th Fibonacci number.… (More)
Rational arithmetic functions are arithmetic functions of the form g 1 * ··· * g r * h −1 1 * ··· * h −1 s , where g i , h j are completely multiplicative functions and * denotes the Dirich-let convolution. Four aspects of these functions are studied. First, some characterizations of such functions are established; second, possible Busche-Ramanujan-type… (More)
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)
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.