Logarithmically complete monotonicity of Catalan-Qi function related to Catalan numbers

In the paper, the authors find the logarithmically complete monotonicity of the Catalan–Qi function related to the Catalan numbers. Subjects: Advanced Mathematics; Analysis Mathematics; Integral Transforms & Equations; Mathematical Analysis; Mathematics & Statistics; Number Theory; Real Functions; Science; Sequences & Series; Special Functions


Introduction
It is stated in Koshy (2009) that the Catalan numbers C n for n ≥ 0 form a sequence of natural numbers that occur in tree enumeration problems such as "In how many ways can a regular n-gon be divided into n − 2 triangles if different orientations are counted separately?" whose solution is the Catalan number C n−2 . The Catalan numbers C n can be generated by C n x n = 1 + x + 2x 2 + 5x 3 + 14x 4 + 42x 5 + 132x 6 + 429x 7 + 1430x 8 + ⋯ .

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The Catalan numbers are a notion in combinatorial science and the theory of numbers. One of their analytic generalizations is the Catalan-Qi function which was introduced by Professor F. Qi and his co-authors in 2015. The set of logarithmically completely monotonic functions, a notion which was explicitly introduced by Professor F. Qi and his co-authors in 2004, is a subset of completely monotonic functions. There is a bijection between the set of completely monotonic functions and the set of the Laplace transforms: a function is a completely monotonic function on the positive semi-axis if and only if it is a Laplace transform of a nonnegative measure. The set of the Stieltjes transforms is a subset of logarithmically completely monotonic function. The reciprocal of a positive Bernstein function is a logarithmically completely monotonic function. In the paper, the authors find the logarithmically complete monotonicity of the Catalan-Qi function.
One of explicit formulas of C n for n ≥ 0 reads that where is the classical Euler gamma function. In Graham, Knuth, and Patashnik (1994), Koshy (2009), and Vardi (1991), it was mentioned that there exists an asymptotic expansion for the Catalan function C x .
A generalization of the Catalan numbers C n was defined in Hilton andPedersen (1991), Klarner (1970), and McCarthy (1992) by for n ≥ 1. The usual Catalan numbers C n = 2 d n are a special case with p = 2.
In combinatorial mathematics and statistics, the Fuss-Catalan numbers A n (p, r) are defined It is obvious that C 1 2 , 2;n = C n , n ≥ 0 and that for a, b, c > 0 and x ≥ 0. In the recent papers of Liu, Shi, and Qi (2015), Mahmoud and Qi (identities), Qi (2015aQi ( , 2015cQi ( , 2015d, Qi, Mahmoud, Shi, and Liu (2015), Qi et al. (2015a), Qi, Shi, and Liu (2015b, 2015c, 2015d, Shi, Liu, and Qi (2015, among other things, some properties, including the general expression and a generalization of the asymptotic expansion (Equation 1), the monotonicity, logarithmic convexity, (logarithmically) complete monotonicity, minimality, Schur-convexity, product and determinantal inequalities, exponential representations, integral representations, a generating function, connections with the Bessel polynomials and the Bell polynomials of the second kind, and identities, of the Catalan numbers C n , the Catalan function C x , the Catalan-Qi function C(a, b; x), and the Fuss-Catalan numbers A n (p, r) were established. Very recently, we discovered in Qi (2015d, Theorem 1.1) a relation between the Fuss-Catalan numbers A n (p, r) and the Catalan-Qi numbers C(a, b; n), which reads that for integers n ≥ 0, p > 1, and r > 0.
From the viewpoint of analysis, motivated by the idea in the papers of Qi and Chen (2007), Qi, Zhang, and Li (2014a, 2014b, 2014c and closely related references cited therein, we will consider in this paper the function and study its properties. Recall from Atanassov and Tsoukrovski (1988), Qi and Chen (2004), Qi and Guo (2004), Schilling, Song, and Vondraček (2012) that an infinitely differentiable and positive function f is said to be logarithmically completely monotonic on an interval I if it satisfies on I for all k ∈ ℕ.
The main results of this paper are the logarithmically complete monotonicity of the function a,b;x (t) in t ∈ [0, ∞) for a, b > 0 and x ≥ 0, which can be stated as the following theorem. (2) the function 1 a,b;x (t) is logarithmically completely monotonic on [0, ∞) if and only if either 0 ≤ x ≤ 1 and a ≥ b or x ≥ 1 and a ≤ b.

Proof of Theorem 1.1
Taking the logarithm of a,b;x (t) and differentiating with respect to t gave Making use of in Abramowitz and Stegun (1972, p. 259, 6.3.21) leads to It is easy to see that the function 1−e −u u is strictly decreasing on (0, ∞). Hence, for u ∈ (0, ∞) if and only if x ⋚ 1. It is apparent that for u ∈ (0, ∞) if and only if a ⋛ b. Recall from Mitrinović, Pečarić, and Fink (1993, Chap. XIII), Schilling et al. (2012, Chap. 1), and Widder (1941 that an infinitely differentiable function f is said to be completely monotonic on an interval I if it satisfies on I for all k ≥ 0. The famous Bernstein-Widder theorem (Widder, 1941, p. 160, Theorem 12a) states that a necessary and sufficient condition that f(x) should be completely monotonic in 0 ≤ x < ∞ is that where is bounded and non-decreasing and the integral (Equation 2) converges for 0 ≤ x < ∞. The proof of Theorem 1.1 is thus complete.
Remark 1 This paper is a slightly modified version of the preprint Qi (2015b).