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)
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 .
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 three recent papers [1-3], solutions of the form x (z) = λz µ are found for iterative functional differential equations. We find similar solutions for a more general equation
The classical folding lemma is extended, in the field of formal series, to two-tier and three-tier folding lemmas covering all possible shapes of the words enclosing one and two middle terms. The twofold and threefold continued fraction identities so obtained are applied to derive a number of explicit continued fractions of certain series expansions,… (More)