A Unified Generalization of the Catalan, Fuss, and Fuss—Catalan Numbers

Feng Qi,Xiao-Ting Shi,Pietro Cerone

doi:10.3390/mca24020049

Abstract

In the paper, the authors introduce a unified generalization of the Catalan numbers, the Fuss numbers, the Fuss–Catalan numbers, and the Catalan–Qi function, and discover some properties of the unified generalization, including a product-ratio expression of the unified generalization in terms of the Catalan–Qi functions, three integral representations of the unified generalization, and the logarithmically complete monotonicity of the second order for a special case of the unified generalization.

Highlights

As well known from [1,2], Catalan numbers Cn are used in the study of set partitions in different areas of mathematics
There are many counting problems in combinatorics whose solution is given by the Catalan numbers Cn
Before getting (7), we did not appreciate the analytic meanings of the form of the product-ratio expression (4) because before catching sight of the unified generalization (5), we did not appreciate the analytic meanings of the form of the Fuss–Catalan numbers An ( p, r ) in (2)

Summary

Introduction

As well known from [1,2], Catalan numbers Cn are used in the study of set partitions in different areas of mathematics. In recent papers [6,22,23,24,25,26,27,28,29,30,31,32], among other things, some properties, including the general expression and a generalization of the asymptotic expansion (1), the monotonicity, logarithmic convexity, (logarithmically) complete monotonicity, minimality, Schur-convexity, product and determinantal inequalities, exponential representations, integral representations, a generating function, connections with the Bessel polynomials and the Bell polynomials of the second kind, and identities, of the Catalan numbers Cn , the Catalan function Cx , the Catalan–Qi numbers C ( a, b; n), the Catalan–Qi function C ( a, b; z), and the Fuss–Catalan numbers An ( p, r ) were established. We will establish the logarithmically complete monotonicity of the second order for the unified generalization

A Unified Generalization of the Catalan and Other Numbers

A Product-Ratio Expression of the Fuss–Catalan–Qi Function

Integral Representations of the Fuss–Catalan–Qi Function

Properties of the Fuss–Catalan–Qi Function

Remarks

Conclusions