We consider the values concerning
$$\mathcal{M}(\theta ,\phi )=\underset{\left|q\right|\to \infty}{lim\; inf}\left|q\right|\left|\right|{q}^{\theta}-\phi \left|\right|$$
where the continued fraction expansion of $\theta $ has a quasi-periodic form. In particular, we treat the cases so that each quasi-periodic form includes no constant. Furthermore, we give some general conditions satisfying $\mathcal{M}(\theta ,\phi )=0$.

Let pₙ/qₙ = [a₀;a₁,...,aₙ] be the n-th convergent of the continued fraction expansion of [a₀;a₁,a₂,...]. 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 studied. In special, recurrence relations and explicit forms of such leaping convergents have been treated. In this paper, we consider recurrence relations and explicit forms of the leaping...

Denote the n-th convergent of the continued fraction by pₙ/qₙ = [a₀;a₁,...,aₙ]. We give some explicit forms of leaping convergents of Tasoev continued fractions. For instance, [0;ua,ua²,ua³,...] is one of the typical types of Tasoev continued fractions. Leaping convergents are of the form ${p}_{rn+i}/{q}_{rn+i}$ (n=0,1,2,...) for fixed integers r ≥ 2 and 0 ≤ i ≤ r-1.

We obtain the values concerning $(\theta ,\varphi )=limin{f}_{\left|q\right|\to \infty}\left|q\right|\Vert q\theta -\varphi \Vert $ using the algorithm by Nishioka, Shiokawa and Tamura. In application, we give the values (θ,1/2), (θ,1/a), (θ,1/√(ab(ab+4))) and so on when θ = (√(ab(ab+4)) - ab)/(2a) = [0;a,b,a,b,...].

Let $\alpha \>1$ be irrational. Several authors studied the numbers
$${\ell}^{m}\left(\alpha \right)=inf\left\{\phantom{\rule{0.166667em}{0ex}}\right|y|:y\in {\Lambda}_{m},\phantom{\rule{0.166667em}{0ex}}y\ne 0\},$$
where $m$ is a positive integer and ${\Lambda}_{m}$ denotes the set of all real numbers of the form $y={\u03f5}_{0}{\alpha}^{n}+{\u03f5}_{1}{\alpha}^{n-1}+\cdots +{\u03f5}_{n-1}\alpha +{\u03f5}_{n}$ with restricted integer coefficients $|{\u03f5}_{i}|\le m$. The value of ${\ell}^{1}\left(\alpha \right)$ was determined for many particular Pisot numbers and ${\ell}^{m}\left(\alpha \right)$ for the golden number. In this paper the value of ${\ell}^{m}\left(\alpha \right)$ is determined for irrational numbers $\alpha $, satisfying ${\alpha}^{2}=a\alpha \pm 1$ with a positive integer $a$.

Many new types of Hurwitz continued fractions have been studied by the author. In this paper we show that all of these closed forms can be expressed by using confluent hypergeometric functions ${}_{0}{F}_{1}(;c;z)$. In the application we study some new Hurwitz continued fractions whose closed form can be expressed by using confluent hypergeometric functions.

We compute upper and lower bounds for the approximation of hyperbolic functions at points $1/s$
$(s=1,2,\cdots )$ by rationals $x/y$, such that $x,y$ satisfy a quadratic equation. For instance, all positive integers $x,y$ with $y\equiv 0\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}2)$ solving the Pythagorean equation ${x}^{2}+{y}^{2}={z}^{2}$ satisfy
$$|ysinh(1/s)-x|\phantom{\rule{0.166667em}{0ex}}\gg \frac{loglogy}{logy}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}.$$
Conversely, for every $s=1,2,\cdots $ there are infinitely many coprime integers $x,y$, such that
$$|ysinh(1/s)-x|\phantom{\rule{0.166667em}{0ex}}\ll \frac{loglogy}{logy}$$
and ${x}^{2}+{y}^{2}={z}^{2}$ hold simultaneously for some integer $z$. A generalization to the approximation of $h\left({e}^{1/s}\right)$ for rational functions $h\left(t\right)$...

We give a generalization of poly-Cauchy polynomials and investigate their arithmetical and combinatorial properties. We also study the zeta functions which interpolate the generalized poly-Cauchy polynomials.

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