Multiple-correction and continued fraction approximation

Abstract

The main aim of this paper is to further develop a multiple-correction method formulated in a previous work [6]. As applications, we find a kind of hybrid-type finite continued fraction approximations in two cases of Landau constants and Lebesgue constants. In addition, we refine the previous results of Lu [29] and Xu and You [48] for the Euler-Mascheroni constant.

Continued fraction formulae involving ratios of three gamma functions

“…the Euler-Mascheroni constant, the constants of Landau, the constants of Lebesgue, etc. (see [9]). However, in some cases the previous two approaches are not very efficient, for instance, the ratios of the gamma functions.…”

“…A large part of this section is taken from [11] which, in turn, follows from earlier works of Mortici [30] and Cao [9]. Let f (x) be a real function defined on (x 0 , +∞) to be approximated with lim x→+∞ f (x) = 0.…”

“…Mortici [30] established a very useful tool for measuring the rate of convergence, which claims that a sequence {x n } ∞ n=1 converging to zero is the fastest possible when the difference {x n − x n+1 } ∞ n=1 is the fastest possible. Since then, the Mortici lemma has been effectively applied in many papers such as [4,9,10,11,12,31]. In fact, the Mortici lemma may be viewed as a special form of Stolz formula.…”

Some new quicker convergences to Glaisher–Kinkelin’s and Bendersky–Adamchik’s constants

“…Motivated by the important work of Mortici [2] and Lu [1], in this paper we will continue our previous works [7][8][9][10], and apply the multiple-correction method to construct some new convergent sequences for Glaisher-Kinkelin's and BenderskyAdamchik's constants, which have faster rate of convergence. Moreover, we establish sharp bounds for the corresponding error terms.…”

Improved Approximation Formulas and Inequalities for the Wallis Ratio

“…Motivated by these works, in this paper we will apply the multiple-correction method [7,8,9] to construct an improved continued fraction asymptotic expansion for the Wallis ratio as follows:…”