title: Some properties of the Catalan-Qi function related to the Catalan numbers.
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.
authors.affiliation: Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, 454010 Henan Province China ; College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, 028043 Inner Mongolia Autonomous Region China ; Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City, 300387 China.
authors.forename: Feng
authors.identifier: 0000-0001-6239-2968
authors.initials: F
authors.lastname: Qi
authors.affiliation: Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516 Egypt.
authors.forename: Mansour
authors.initials: M
authors.lastname: Mahmoud
authors.affiliation: Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City, 300387 China.
authors.forename: Xiao-Ting
authors.initials: XT
authors.lastname: Shi
authors.affiliation: Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City, 300387 China.
authors.forename: Fang-Fang
authors.initials: FF
authors.lastname: Liu
keywords: Asymptotic expansion; Catalan function; Catalan number; Catalan–Qi function; Complete monotonicity; Inequality; Integral representation; Logarithmic convexity; Logarithmically complete monotonicity; Minimality; Property; Ratio of gamma functions