Multiple-correction and continued fraction approximation (II)

Abstract

The main aim of this paper is to further develop a multiple-correction method formulated in a previous work [6]. As its 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 [44] for the Euler-Mascheroni constant.

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.…”

Multiple-correction and summation of the rational series

“…(Step 1) Let us develop further the previous multiple-correction method formulated in [6]. In fact, the multiple-correction method is a recursive algorithm, and one of its advantages is that by repeating correction-process we always can accelerate the rate of approximation.…”

“…(e.g. see [6]). At the same times, it predicts that finding a continued fraction representation started started from this series seems "hopeless", one should replace other series expressions to try again and again, or find a linear combination solution of several continued fractions.…”

“…In Sec. 3, based on the work in [6], we shall formulate a systematical method to look for either a simple closed form solution or the fastest possible finite continued fraction approximation solution for a class of linear difference equation of order one, sometimes we may guess further its continued fraction solution. In order to prove our new "conjectures", in Sec.…”

Approximation for the gamma function via the tri-gamma function

“…In this paper we will apply the multiple-correction method [3–5] to construct a new asymptotic expansion for the factorial n! and the gamma function via the tri-gamma function.…”