The continued fraction expansion of Euler's constant

Beta-expansion and continued fraction expansion over formal Laurent series

Let x ∈ I be an irrational element and n 1, where I is the unit disc in the field of formal Laurent series F((X−1)), we denote by kn(x) the number of exact partial quotients in continued fraction expansion of x, given by the first n digits in the β-expansion of x, both expansions are based on F((X−1)). We obtain that lim inf n→+∞ kn(x) n = degβ 2Q∗(x) , lim sup n→+∞ kn(x) n = degβ 2Q∗(x) , wher...

A Short Proof of the Simple Continued Fraction Expansion of

One of the most interesting proofs is due to Hermite; it arose as a byproduct of his proof of the transcendence of e in [5]. (See [6] for an exposition by Olds.) The purpose of this note is to present an especially short and direct variant of Hermite’s proof and to explain some of the motivation behind it. Consider any continued fraction [a0, a1, a2, . . .]. Its ith convergent is defined to be ...

On the Continued Fraction Expansion of a Class of Numbers

(a general reference is Chapter I of [9]). If ξ is irrational, then, by letting X tend to infinity, this provides infinitely many rational numbers x1/x0 with |ξ − x1/x0| ≤ x 0 . By contrast, an irrational real number ξ is said to be badly approximable if there exists a constant c1 > 0 such that |ξ − p/q| > c1q for each p/q ∈ Q or, equivalently, if ξ has bounded partial quotients in its continue...

A Proof of the Continued Fraction Expansion of 1/M

Ramanujan and the Regular Continued Fraction Expansion of Real Numbers

In some recent papers, the authors considered regular continued fractions of the form [a0; a, · · · , a } {{ } m , a, · · · , a } {{ } m , a, · · · , a } {{ } m , · · · ], where a0 ≥ 0, a ≥ 2 and m ≥ 1 are integers. The limits of such continued fractions, for general a and in the cases m = 1 and m = 2, were given as ratios of certain infinite series. However, these formulae can be derived from ...