Integral Representations of the Catalan Numbers and Their Applications

Abstract

In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several power series whose coefficients involve the Catalan numbers.

Keywords

• Catalan number
• integral representation
• equivalent relation
• application
• sum of power series
• Catalan–Qi function
• Catalan–Qi number
• beta function