Logarithmically complete monotonicity of Catalan-Qi function related to Catalan numbers

Feng Qi,Bai-Ni Guo,Lishan Liu

doi:10.1080/23311835.2016.1179379

Abstract

In the paper, the author finds the logarithmically complete monotonicity of the Catalan–Qi function related to the Catalan numbers.

Highlights

It is stated in Koshy (2009) that the Catalan numbers Cn for n ≥ 0 form a sequence of natural numbers that occur in tree enumeration problems such as “In how many ways can a regular n-gon be divided into n − 2 triangles if different orientations are counted separately?” whose solution is the Catalan number Cn−2
We discovered in Qi (2015d, Theorem 1.1) a relation between the Fuss–Catalan numbers An(p, r) and the Catalan–Qi numbers C(a, b; n), which reads that
The main results of this paper are the logarithmically complete monotonicity of the function a,b;x(t) in t ∈ [0, ∞) for a, b > 0 and x ≥ 0, which can be stated as the following theorem

Summary

Introduction

It is stated in Koshy (2009) that the Catalan numbers Cn for n ≥ 0 form a sequence of natural numbers that occur in tree enumeration problems such as “In how many ways can a regular n-gon be divided into n − 2 triangles if different orientations are counted separately?” whose solution is the Catalan number Cn−2.

PUBLIC INTEREST STATEMENT

It is easy to see that r

Findings

It is obvious that