The Catalan-Qi number of the second kind and a related integral

TL;DR

This paper derives a new representation of the Catalan-Qi number of the second kind using hypergeometric functions, Legendre polynomials, and other special functions, expanding the analytical tools available for these numbers.

Contribution

It introduces a novel reduction of the integral formula for the Catalan-Qi number to a product involving hypergeometric and Legendre functions, linking it to various special functions.

Findings

Integral expressed as a product of Catalan number and hypergeometric function

Hypergeometric function related to associated Legendre polynomial

Connections established with Beta function and Gegenbauer polynomials

Abstract

Li et al. give an integral formula for the Catalan-Qi number of the second kind. They show that this integral can be written as a summation with double factorials. In this paper the integral is reduced to a product of the Catalan number and a hypergeometric function. This hypergeometric function can be written as an associated Legendre polynomial of the first kind. There are also connections with the incomplete Beta function and the Gegenbauer polynomials. In the last section a related integral is calculated.