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)
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)
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)
A factorization theorem is proved for a class of generalized exponential polynomials having all but finitely many of integer zeros belong to a finite union of arithmetic progressions. This theorem extends a similar result for ordinary exponential polynomials due to H. N. Shapiro in 1959. The factorization makes apparent those factors corresponding to all… (More)
A generalized Euler's totient is defined as a Dirichlet convolution of a power function and a product of the Souriau-Hsu-M ¨ obius function with a completely multiplicative function. Two combinatorial aspects of the generalized Euler's totient, namely, its connections to other totients and its relations with counting formulae, are investigated.
Recommended by Stéphane Louboutin Two kinds of series representations, referred to as the Engel series and the Cohen-Egyptian fraction expansions, of elements in two different fields, namely, the real number and the discrete-valued non-archimedean fields are constructed. Both representations are shown to be identical in all cases except the case of real… (More)