A further generalization of the Catalan numbers and its explicit formula and integral representation

Abstract

In the paper, motivated by the generating function of the Catalan numbers in combinatorial number theory and with the aid of Cauchy's integral formula in complex analysis, the authors generalize the Catalan numbers and its generating function, establish an explicit formula and an integral representation for the generalization of the Catalan numbers and corresponding generating function, and derive several integral formulas and combinatorial identities.

Integral representations, monotonic properties, and inequalities on extended central binomial coefficient

“…In Dana-Picard and Zeitoun, 8, Corollary 3.2 Penson and Sixdeniers, 9, p. 2, Eq. (10) Li et al, 10, Section 4.2 and Qi, 8, Theorem 3.1 among other things, the integral representations…”

“…Remark The proof of Theorem 2.1 provides an alternative proof of the integral representation of the Catalan numbers for n ≥0. For details on the sequence of the Catalan numbers C n , please refer to the papers 8,10,19,20 . Several integral representations of the Catalan numbers C n have been reviewed and surveyed in the paper 12 …”

Several combinatorial identities derived from series expansions of powers of arcsine

“…in combinatorial number theory have attracted many mathematicians who have published several monographs [18,22,41,45] and a number of papers such as [24,25,27,28,35,36,37,38,39,40].…”

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

“…There are many generalizations of the Catalan numbers. Following the previous formula it is a natural generalization to take the generation function [6], [5]…”

“…We get C n (1/2, 1/4) = C n . The authors in [6] call this generalization the Catalan-Qi numbers of the second kind. They derive a formula with a summation with double factorials for these numbers [6, 2.1]…”