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.
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.