Some 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, and inequalities of the Catalan numbers, the Catalan function, and the Catalan–Qi function. As by-products, an exponential expansion and a double inequality for the ratio of two gamma functions are derived.


Background
It is stated in Koshy (2009), Stanley and Weisstein (2015) 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 Two of explicit formulas of C n for n ≥ 0 read that where (1) C n x n = 1 + x + 2x 2 + 5x 3 + 14x 4 + 42x 5 + 132x 6 + 429x 7 + 1430x 8 + · · · .
In Qi et al. (2015b, Remark 1) an analytical generalization of the Catalan numbers C n and the Catalan function C x was given by and the integral representation for a, b > 0 and x ≥ 0 was derived. For uniqueness and convenience of referring to the quantity (6), we call C(a, b; x) the Catalan-Qi function. It is clear that The integral representation (7) generalizes an integral representation for C 1 2 , 2; x in Shi et al. (2015). Currently we do not know and understand the combinatorial interpretations of C(a, b; x) and its integral representation (7). Here we would not like to discuss the combinatorial interpretations of them. What we concern here is the asymptotic expansion similar to (5) for C (a, b; x). p F q (a 1 , . . . , a p ; b 1 , . . . , b q ; z) = ∞ n=0 (a 1 ) n · · · (a p ) n (b 1 ) n · · · (b q ) n z n n!
From the power series (1), we observe that the Catalan numbers C n is an increasing sequence in n ≥ 0 with C 0 = C 1 . What about the monotonicity and convexity of the Catalan numbers C n , the Catalan function C x , and the Catalan-Qi function C(a, b; x)? In Temme (1996, p. 67), it was listed that Accordingly, we obtain an alternative integral representation for b > a > 0 and x ≥ 0, where B(z, w) denotes the classical beta function which can be defined (Abramowitz and Stegun 1972, p. 258, 6.2.1 and 6.2.2) by for R(z) > 0 and R(w) > 0 and satisfies From the integral representations (8) and (9), one can not apparently see any message about the monotonicity and convexity of the Catalan-Qi function C (a, b; x) As showed by (1), the Catalan numbers C n have a generating function 2 1+ √ 1−4x . What is the generating function of the Catalan-Qi numbers C(a, b; n)?
The aim of this paper is to supply answers to the above problems and others. (8)

A new expression of the Catalan numbers
In order to establish a new expression for the Catalan numbers C n , we need the following lemma which was summarized up in the papers Qi (2015c, Section 2.2, p. 849), Qi (2016, p. 94), and Wei and Qi (2015, Lemma 2.1) from Bourbaki (2004, p. 40, Exercise 5).
for k ∈ N as x → 0, making use of the formula (13) yields By virtue of the second function in the Eq. (1), we see that The proof of Theorem 1 is complete.

Asymptotic expansions of the Catalan-Qi function C(a, b; x)
We first derive two asymptotic expansions of the Catalan-Qi function C(a, b; x). Consequently, from these two asymptotic expansions, we deduce a general expression for (5) and an asymptotic expansion of the ratio Ŵ(a) Ŵ(b) for a, b > 0.
e xz z e z − 1 For b > a > 0, the Catalan-Qi function C(a, b; x) has the asymptotic expansion as x → ∞. Consequently, the Catalan function C x has the asymptotic expansion as x → ∞.
Proof In Temme (1996, p. 67), it was listed that, under the condition in the sector | arg z| < π, where the generalized Bernoulli polynomials B (σ ) k (x) are defined by (14) in Temme (1996, p. 4). Consequently, the function C(a, b; x) has the asymptotic expansion (15) under the condition b > a > 0 as x → ∞. In particular, when taking a = 1 2 and b = 2 in (15), we obtain the asymptotic expansion (16). Theorem 2 is thus proved.
Remark 1 In Qi (2015a), there are another two asymptotic expansions for C n and C x , which were established by virtue of the integral representations (8) and (7) for a = 1 2 and b = 2. (16) is a general expression of the asymptotic expansion (5). Hence, the asymptotic expansion (15) is a generalization of (5).

Theorem 3 Let B i denote the Bernoulli numbers defined by
Then the Catalan-Qi function C(a, b; x) has the exponential expansion where I(α, β) denotes the exponential mean defined by Proof Making use of (17) in the integral representation (7) yields which can be reformulated as the form (18). The exponential expansion (20) follows from letting x → 0 in (18) and rearranging. Theorem 3 is thus proved.
Remark 4 For more information on the exponential mean I(α, β) in (19), please refer to the monograph (Bullen 2003) and the papers Qi 2009, 2011).

Integral representations and complete monotonicity of the Catalan-Qi function C(a, b; x)
Motivated by the first integral representations (8) and (9), we guess out the following integral representations for the Catalan-Qi function C(a, b; x).
Theorem 4 For b > a > 0 and x ≥ 0, the Catalan-Qi function C(a, b; x) has integral representations (20) and Proof Straightforwardly computing and directly utilizing (11) and (12) acquire The integral representation (21) is thus proved. Similar to the above argument, by virtue of (11) and (12), we obtain Hence, the integral representation (22) follows readily. The proof of Theorem 4 is thus complete.

Logarithmically complete monotonicity of the Catalan-Qi function C(a, b; x)
An infinitely differentiable and positive function f is said to be logarithmically completely monotonic on an interval The inclusions were discovered in Berg (2004), , Qi and Chen (2004), Qi and Guo (2004) Recall from monographs Mitrinović et al. (1993, pp. 372-373) and Widder (1941, p. 108, Definition 4) that a sequence {µ n } 0≤n≤∞ is said to be completely monotonic if its elements are non-negative and its successive differences are alternatively nonnegative, that is, for n, k ≥ 0, where Recall from Widder (1941, p. 163, Definition 14a) that a completely monotonic sequence {a n } n≥0 is minimal if it ceases to be completely monotonic when a 0 is decreased.
Proof In Qi and Li (2015, Theorem 1.1), it was proved that, when a ≷ b, the function for c > 0 is logarithmically completely monotonic on [0, ∞) if and only if c Ŵ(b) Ŵ(a) . It is easy to see that Therefore, the function C ±1 (a, b; x) is logarithmically completely monotonic on [0, ∞) if and only if a ≷ b. Consequently, the function C −1 1 2 , 2; x is logarithmically completely monotonic, and then completely monotonic and logarithmically convex, on [0, ∞) . As a result, the complete monotonicity, minimality, and logarithmic convexity of the sequence (27) follows immediately from Widder (1941, p. 164, Theorem 14b) which reads that a necessary and sufficient condition that there should exist a completely monotonic function f(x) in 0 ≤ x < ∞ such that f (n) = a n for n ≥ 0 is that {a n } ∞ 0 should be a minimal completely monotonic sequence. The proof of Theorem 6 is complete.
Remark 8 It is interesting that, since the function h a,b;c (x) defined by (28) originates from the coding gain (see Lee and Tepedelenlioğlu 2011;Qi and Li 2015), Theorem 6 and its proof imply some connections and relations among the Catalan numbers, the coding gain, and the ratio of two gamma functions.
Theorem 7 Let a, b > 0 and x ≥ 0. Then 1. when b > a, the function C(a, b; x) is decreasing in x ∈ [0, x 0 ), increasing in x ∈ (x 0 , ∞),and logarithmically convex in x ∈ [0, ∞); 2. when b < a, the function C(a, b; x) is increasing in x ∈ [0, x 0 ), decreasing in x ∈ (x 0 , ∞), and logarithmically concave in x ∈ [0, ∞); where x 0 is the unique zero of the equation and satisfies x 0 ∈ 0, 1 2 . Consequently, the Catalan numbers C n for n ∈ N is strictly increasing and logarithmically convex.
Proof In Guo and Qi (2010, Theorem 1) closely-related references therein, it was proved that the function is completely monotonic on (0, ∞) if and only if α ≤ 1. This means that  (30) and (31), we see that the Catalan-Qi function C(a, b; x) for all a, b > 0 with a � = b is not monotonic on [0, ∞) and that 1. when b > a, the function C(a, b; x) is decreasing in x ∈ (0, x 0 ) and increasing in x ∈ (x 0 , ∞); 2. when b < a, the function C (a, b; x) is increasing in x ∈ (0, x 0 ) and decreasing in x ∈ (x 0 , ∞); where x 0 is the unique zero of the Eq. (29). The Eq. (29) can be rearranged as Regarding b as a variable and differentiating with respect to b give which can be reformulated as Employing the asymptotic expansion in Abramowitz and Stegun (1972, p. 260, 6.4.11) yields Due to [ψ ′ (x)] 2 + ψ ′′ (x) > 0 on (0, ∞), see Alzer (2004), Qi (2015b), Qi and Li (2015), Qi et al. (2013) and plenty of closely-related references therein, the function u − 1 ψ ′ (u) is strictly increasing, and so on (0, ∞). Accordingly, the unique zero x 0 of the Eq. (29) belongs to 0, 1 2 . It is immediate that Since the tri-gamma function ψ ′ (x) is completely monotonic on (0, ∞), inequalities for k ∈ N hold if and only if b ≶ a. The proof of Theorem 7 is complete.
Remark 9 From Theorem 7, we can derive that, for b > a > 0, In other words, Theorem 8 For b > a > 0, the function is logarithmically completely monotonic on [0, ∞).
Proof By (6), it follows that which can be straightforwardly verified to be a logarithmically completely monotonic function on [0, ∞). By the first inclusion in (26), we obtain the required complete monotonicity of the function (32).
Remark 10 The integral representation (22) can be rewritten as for b > a > 0 and x ≥ 0. This formula and both of the integral representations (10) and (25) all mean that the function (32) for b > a > 0 is completely monotonic on [0, ∞) . This conclusion is weaker than Theorem 8.
Proof This follows from the integral representation (7).
Remark 11 Theorems 8 and 9 imply that the sequences are logarithmically completely monotonic and minimal, which have been concluded in Qi (2015a, Theorems 1.1 and 1.2).
C n 4 n n≥0 and (n + 2) n+3/2 (n + 1/2) n C n 4 n n≥0 Qi et al. SpringerPlus (2016) 5:1126 A generating function of the Catalan-Qi sequence C(a, b; n) In this section, we discover that 2 F 1 a, 1; b; bt a is a generating function of the Catalan-Qi numbers C(a, b; n).
Theorem 10 For a, b > 0 and n ≥ 0, the Catalan- Qi numbers C(a, b; n) can be generated by and, conversely, satisfy Proof Using the relation (z) n Ŵ(z) = Ŵ(z + n) for n ≥ 0, we have As a result, we obtain Using the relation (−n) n+i = 0 for i ∈ N, which can be derived from (4), we obtain Further using the relation we acquire The formula (Graham et al. 1994, p. 192, (5.48)) reads that Hence, the inversion of the relation (35) gives us the relation (34). The proof of Theorem 10 is complete. (−1) r (−n) r r! C(a, b; r).

.1), it is collected that
In order to prove the Eq. (33), it is sufficient to show In fact, a straightforward calculation reveals for b > 1. This gives an alternative proof of (33) for b > 1.