A logarithmically completely monotonic function involving the gamma function and originating from the Catalan numbers and function

Abstract

In the paper, the authors find necessary conditions and sufficient conditions for a function involving the gamma function and originating from investigation of properties of the Catalan numbers and function in combinatorics to be logarithmically completely monotonic.

Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

“…For more information on the Catalan numbers C k and their recent developments, please refer to the monographs [2,12,55], the papers [13,25,34,40,46,47,54,59,61,62,63] and the closely related references therein. The first six Chebyshev polynomials of the second kind U k (x) for 0 ≤ k ≤ 5 are U 0 (x) = 1, U 1 (x) = 2x, U 2 (x) = 4x 2 − 1, U 3 (x) = 8x 3 − 4x, U 4 (x) = 16x 4 − 12x 2 + 1, U 5 (x) = 32x 5 − 32x 3 + 6x.…”

Schur-convexity of the Catalan–Qi function related to the Catalan numbers

“…This paper is a companion of the articles [8,9,13,15,16,19,24,26,27,28,33] and a slightly revised version of the preprint [29].…”

“…Here we would not like to discuss its combinatorial interpretation. Recently, in the papers [8,13,19,20,24,26,27,28,33], the authors presented asymptotic expansions, integral representations, logarithmic convexity, complete monotonicity, minimality, logarithmically complete monotonicity, a generating function, and inequalities of the Catalan numbers C n , the Catalan function C x , the Catalan-Qi function C(a, b; x) in x, and related functions involving the ratios…”

An Improper Integral with a Square Root

“…In recent years, the Catalan numbers C n has been analytically generalized and studied in [4,5,9,10,13,15,16,17] and closely-related references therein.…”