Integral Representations of Catalan Numbers and Their Applications

Abstract

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.

Reconfigurable Integrated Optical Interferometer Network-Based Physically Unclonable Function

“…To overcome the configuration limit set by the standard Catalan numbers, we must utilize a lesser-known combinatorics counting method for binary trees as described by F. Qi et al [21], the method of counting by integral representation of the Catalan numbers. The method of integral counting can be directly applied to the planar tree variation of counting, similar to the work by P. Flajolet et al in [22].…”

Catalan–Qi Numbers, Series Involving the Catalan–Qi Numbers and a Hankel Determinant Evaluation

“…)), and this coefficient can be generalized to all complex s ≠ − n, t ≠ − n, and s − t ≠ − n by the following formula: [8][9][10][11][12]. is generalization of the Catalan numbers C n was introduced according to Euler-Gamma function as follows:…”

Catalan-like number sequences and Hausdorff moment sequences

“…respectively. For more information about integral representations of the Catalan numbers, see [19] and references therein.…”

“…Finding integral representations of well-known counting numbers has been studied( [19] and reference therein). Finding a correct interval of the support of the representing measure will help one find an integral representations.…”