Parametric integrals, the Catalan numbers, and the beta function
type: Item
instance of: scholarly article
title: Parametric integrals, the Catalan numbers, and the beta function
zbMATH Open document ID: 1440.11022
DOI: 10.4171/EM/332
author: Feng Qi
published in: Elemente der Mathematik
publication date: 2017-11-17T00:00:00Z
review text: Summary: In mathematics, computing definite integrals and improper integrals is important. The beta and gamma functions, also called the Euler integrals of the first and second kinds, can be defined by definite (improper) integrals and are most extensively-applied special functions. The Catalan numbers form a sequence of natural numbers that occur in various counting problems in combinatorial mathematics. The Wallis ratio is connected with the ratio of two gamma functions, with the ratio of two double factorials, and with definite integral whose integrand is the power of sine or cosine and has applications in statistics. In the paper, starting from a sequence of parametric integrals, the author presents a unified expression of two known results on the sequence of parametric integrals in terms of the gamma function, computes a new sequence of parametric integrals in terms of the gamma function, finds the absolute convexity of the Catalan numbers, calculates a general sequence of parametric integrals in terms of the beta function, and represents the above parametric integrals, the Catalan numbers, and the Wallis ratio in terms of the beta function.
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 33B15
zbMATH DE Number: 6808302
zbMATH Keywords: Catalan numbers
zbMATH Keywords: beta function
zbMATH Keywords: Wallis ratio
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/em/332
OpenAlex ID: W2738678714
description: scientific article; zbMATH DE number 6808302