The fastest possible continued fraction approximations of a class of functions

TL;DR

This paper develops a systematic method for constructing the fastest continued fraction approximations of certain functions, leading to new expansions, inequalities, and conjectures, with applications to gamma functions and geometric volumes.

Contribution

It introduces a novel systematic approach combining multiple-correction, Mortici's lemma, and transformations to achieve the fastest continued fraction approximations.

Findings

Derived sharp inequalities for specific functions.

Presented the fastest continued fraction expansion of a Ramanujan ratio.

Proposed three new conjectures related to continued fractions.

Abstract

The goal of this paper is to formulate a systematical method for constructing the fastest possible continued fraction approximations of a class of functions. The main tools are the multiple-correction method, the generalized Mortici's lemma and the Mortici-transformation. As applications, we will present some sharp inequalities, and the continued fraction expansions associated to the volume of the unit ball. In addition, we obtain a new continued fraction expansion of Ramanujan for a ratio of the gamma functions, which is showed to be the fastest possible. Finally, three conjectures are proposed.