Vichian Laohakosol

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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)
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)