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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)

Given a set of positive integers A = {a 1 ,. .. , a n }, we study the number p A (t) of nonnegative integer solutions (m 1 ,. .. , m n) to n j=1 m j a j = t. We derive an explicit formula for the polynomial part of p A. Let A = {a 1 ,. .. , a n } be a set of positive integers with gcd(a 1 ,. .. , a n) = 1. The classical Frobenius problem asks for the… (More)

- Takao Komatsu, Alfred J Van Der Poorten
- 2001

We provide a fairly simple and straightforward argument yielding all substitution invariant Beatty sequences.

The Fibonacci Zeta functions are defined by ζ F (s) = ï¿¿ ∞ k=1 F −s k. Several aspects of the function have been studied. In this article we generalize the results by Ohtsuka and Nakamura, who treated the partial infinite sum ï¿¿ ∞ k=n F −s k for all positive integers n.

In this paper we consider infinite sums derived from the reciprocals of the Fibonacci numbers, and infinite sums derived from the reciprocals of the square of the Fibonacci numbers. Applying the floor function to the reciprocals of these sums, we obtain equalities that involve the Fibonacci numbers.

- Takao Komatsu
- 2006

We obtain the values concerning M(θ, φ) = lim inf |q|→∞ |q||qθ−φ using the algorithm by Nishioka, Shiokawa and Tamura. In application, we give the values M(θ, 1/2), M(θ, 1/a), M(θ, 1/ ab(ab + 4)) and so on when θ = (ab(ab + 4) − ab)/(2a) = [0; a, b, a, b,. . .]. 1. Introduction. Let θ be irrational and φ real. We suppose throughout that qθ − φ is never… (More)

For any given real number, its corresponding continued fraction is unique. However, given an arbitrary continued fraction, there has been no general way to identify its corresponding real number. In this paper we shall show a general algorithm from continued fractions to real numbers via infinite sums representations. Using this algorithm, we obtain some… (More)

- Ken Kamano, Takao Komatsu
- 2013

We introduce the poly-Cauchy polynomials which generalize the classical Cauchy polynomials, and investigate their arithmetical and combinatorial properties. These polynomials are considered as analogues of the poly-Bernoulli polynomials that generalize the classical Bernoulli polynomials. Moreover, we investigate the zeta functions which interpolate the… (More)

- Takao Komatsu
- 2013

Let p n /q n = [a 0 ; a 1 ,. .. , a n ] be the n-th convergent of the continued fraction expansion of [a 0 ; a 1 , a 2 ,. .. ]. Leaping convergents are those of every r-th convergent p rn+i /q rn+i (n = 0, 1, 2,. . .) for fixed integers r and i with r ≥ 2 and i = 0, 1,. .. , r − 1. The leaping convergents for the e-type Hurwitz continued fractions have been… (More)