Some properties of the Catalan-Qi function related to the Catalan numbers.

Feng Qi,Xiao-Ting Shi,Mansour Mahmoud,Fang-Fang Liu

doi:10.1186/s40064-016-2793-1

Abstract

In the paper, the authors find some properties of the Catalan numbers, the Catalan function, and the Catalan–Qi function which is a generalization of the Catalan numbers. Concretely speaking, the authors present a new expression, asymptotic expansions, integral representations, logarithmic convexity, complete monotonicity, minimality, logarithmically complete monotonicity, a generating function, and inequalities of the Catalan numbers, the Catalan function, and the Catalan–Qi function. As by-products, an exponential expansion and a double inequality for the ratio of two gamma functions are derived.

Highlights

Mathematics Subject Classification: Primary 33B15; Secondary 05A10, 05A15, 05A16, 05A19, 05A20, 11B83, 11Y55, 11Y60, 26A48, 26A51, 26D15, 33C05, 44A20. It is stated in Koshy (2009), Stanley and Weisstein (2015) 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
The Catalan numbers Cn can be generated by 2 √ 1 + 1 − 4x
A double inequality of the Catalan–Qi function C(a, b; x) we present a double inequality of the Catalan–Qi function C(a, b; x)

Summary

Introduction

It is stated in Koshy (2009), Stanley and Weisstein (2015) 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. The Catalan numbers Cn can be generated by 2 √ 1 + 1 − 4x √ = 1− 1 − 4x 2x ∞ = Cnxn n=0. Two of explicit formulas of Cn for n ≥ 0 read that 4nŴ(n + 1/2) Cn =

Objectives

Conclusion