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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)
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)
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)