Xiao-Ting Shi

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

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.

Several identities involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers

Abstract In the paper, the authors find several identities, including a new recurrence relation for the Stirling numbers of the first kind, involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers.

A double inequality for an integral mean in terms of the exponential and logarithmic means

In the paper, the authors aim to present a double inequality for the integral mean

Some properties of central Delannoy numbers

\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }a^{\cos ^2\theta }b^{\sin ^2\theta }{{\mathrm{d}}}\theta \end{aligned}

An integral representation of the Catalan numbers

12π∫02πacos2θbsin2θdθin terms of the exponential and logarithmic means. For attaining the goal, by the Cauchy residue theorem in the theory of complex functions and properties of definite integrals, the authors represent the above integral mean in terms of the modified Bessel function of the first kind. Finally, by virtue of inequalities for the hyperbolic tangent function, the authors further refine upper bounds in the newly-established double inequality in terms of the arithmetic and geometric means.

A logarithmically completely monotonic function involving the gamma function and originating from the Catalan numbers and function

Abstract In the paper, the authors present the Schur-convexity of the absolute of the logarithm of the Catalan-Qi function and prove the logarithmically complete monotonicity of the Catalan-Qi function.

Expansions of the exponential and the logarithm of power series and applications

Abstract In the paper, by investigating the generating function of central Delannoy numbers, the authors establish several explicit expressions, including determinantal expressions, for central Delannoy numbers, present three identities involving the Cauchy products of central Delannoy numbers, discover an integral representation for central Delannoy numbers, find (absolute) monotonicity, convexity, and logarithmic convexity for the sequence of central Delannoy numbers, and construct several product and determinantal inequalities for central Delannoy numbers.