Integral Representations of the Catalan Numbers and Their Applications
نویسندگان
چکیده
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.
منابع مشابه
Integral Representations of Catalan Numbers and Their Applications
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.
متن کاملTwo Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind
Riordan matrix methods and properties of generating functions are used to prove that the entries of two Catalan-type Riordan arrays are linked to the Chebyshev polynomials of the first kind. The connections are that the rows of the arrays are used to expand the monomials (1/2) (2x) and (1/2) (4x) in terms of certain Chebyshev polynomials of degree n. In addition, we find new integral representa...
متن کاملAn Improper Integral with a Square Root
In the paper, the author presents explicit and unified expressions for a sequence of improper integrals in terms of the beta functions and the Wallis ratios. Hereafter, the author derives integral representations for the Catalan numbers originating from combinatorics.
متن کاملSome properties of the Catalan–Qi function related to the Catalan numbers
In the paper, the authors find some properties of the Catalan numbers, the Catalan function, and the Catalan-Qi function which is a generalization of the Catalan numbers. Concretely speaking, the authors present a new expression, asymptotic expansions, integral representations, logarithmic convexity, complete monotonicity, minimality, logarithmically complete monotonicity, a generating function...
متن کاملOn the Integral Representations of Generalized Relative Type and Generalized Relative Weak Type of Entire Functions
In this paper we wish to establish the integral representations of generalized relative type and generalized relative weak type as introduced by Datta et al [9]. We also investigate their equivalence relation under some certain conditions.
متن کامل