An analytic generalization of the Catalan numbers and its integral representation

In the paper, the authors analytically generalize the Catalan numbers in combinatorial number theory, establish an integral representation of the analytic generalization of the Catalan numbers by virtue of Cauchy's integral formula in the theory of complex functions, and point out potential directions to further study.

The Catalan numbers C n for n ≥ 0 have several integral representations which have been surveyed in [27,Section 2]. The integral representation was discovered in [18] and applied in [47]. An alternative integral representation was derived from the integral representation which was established in [40,Theorem 1.3] by virtue of Cauchy's integral formula in the theory of complex functions. The generalized Catalan function C(a, b; z) defined by (1.3) has also several integral representations which have been surveyed in [27,Section 2]. For example, corresponding to integral representations in (1.5) and (1.6), integral representations (1.9) for b > a > 0 and x ≥ 0 were established in [35,Theorem 4], where the classical beta function B(z, w) can be defined or expressed [21,54] by for (z), (w) > 0. We note that, when letting a = 1 2 and b = 2, the integral representations (1.8) and (1.9) become those in (1.5) and (1.6) respectively.
The generating function G(x) in (1.2) can be regarded as a special case a = 1 2 , b = 1 4 , and c = 1 of the function Essentially, it is better to regard the function as a generalization of the generating function G(x), because but we can not express G a,b (x) in terms of G(x). Now we would like to pose the following three problems.
(1) Can one establish an explicit formula for the sequence C n (a, b) generated by for a ≥ 0 and b > 0? (2) Can one find an integral representation for the sequence C n (a, b) by finding an integral representation of the generating function G a,b (x) in (1.10)? (3) Can one combinatorially interpret the sequence C n (a, b) or some special case of C n (a, b) except the case a = 1 2 and b = 1 4 ? It is easy to see that for n ≥ 0, where the notation for α = 0 is called the falling factorial [36,37,41]. Comparing (1.13) with (1.4) reveals that C(a, b; n) ≡ C n (a, b), although it is possible that or that there exist two 2-tuples (a n , b n ) ∈ (0, ∞) × (0, ∞) and (α n , β n ) ∈ (0, ∞) × (0, ∞) such that C(a n , b n ; n) = C n (α n , β n ) for all n ≥ 0. For our own convenience and referencing to the convention in mathematical community, while calling C(a, b; n) for n ≥ 0, a ≥ 0, and b > 0 generalized Catalan numbers of the first kind, we call C n (a, b) for n ≥ 0, a ≥ 0, and b > 0 generalized Catalan numbers of the second kind.
In this paper, we will give solutions to the first two problems above: establishing an explicit formula for generalized Catalan numbers of the second kind C n (a, b) and finding an integral representation for generalized Catalan numbers of the second kind C n (a, b) by finding an integral representation of the generating function G a,b (x) in (1.10), while leaving the third problem above to interested combinatorists.

An explicit formula for generalized Catalan numbers of the second kind
In this section, we will establish an explicit formula for generalized Catalan numbers of the second kind C n (a, b), which gives a solution to the first problem posed on page 3.
Theorem 2.1. The generalized Catalan numbers of the second kind C n (a, b) for n ≥ 0, a ≥ 0, and b > 0 can be explicitly computed by where the double factorial of negative odd integers −(2 + 1) is defined by Proof. The Bell polynomials of the second kind B n,k ( The famous Faà di Bruno formula can be described [5, p. 139, Theorem C] in terms of the Bell polynomials of the second kind B n,k ( where f • h denotes the composite of the n-time differentiable functions f and h.
where we used the formulas , which can be rearranged as (2.1). The proof of Theorem 2.1 is complete. Stimulated by these two identities and the formula (2.1) in Theorem 2.1, we would like to ask a question: can one use a simple quantity to express the sum 3. An integral representation for generalized Catalan numbers of the second kind In this section, we will find an integral representation for generalized Catalan numbers of the second kind C n (a, b) by finding an integral representation of the generating function G a,b (x) in (1.10), which gives a solution to the second problem posed on page 3.
Theorem 3.1. The principal branch of the generating function G a,b (z) for a ≥ 0 and b > 0 can be represented by Consequently, generalized Catalan numbers of the second kind C n (a, b) for a ≥ 0 and b > 0 can be represented by where i = √ −1 is the imaginary unit and arg z stands for the principal value of the argument of z. By virtue of Cauchy's integral formula [6, p. 113] in the theory of complex functions, for any fixed where L is a positively oriented contour L(r, R) in C \ [0, ∞), as showed in Figure 1, satisfying (1) 0 < r < |z 0 | < R; (2) L(r, R) consists of the half circle z = re iθ for θ ∈ π 2 , 3π 2 ; (3) L(r, R) consists of the line segments z = x ± ir for x ∈ (0, R(r)], where R(r) = √ R 2 − r 2 ; (4) L(r, R) consists of the circular arc z = Re iθ for θ ∈ arctan r R(r) , 2π − arctan r R(r) ; (5) the line segments z = x ± ir for x ∈ (0, R(r)] cut the circle |z| = R at the points R(r) ± ir and R(r) → R as r → 0 + . The integral on the circular arc z = Re iθ equals 1 2πi  The integral on the half circle z = re iθ for θ ∈ π 2 , 3π Consequently, it follows that 1 a + exp ln(−z0) for any z 0 ∈ C \ [0, ∞) and arg z 0 ∈ (0, 2π). Due to the point z 0 in (3.3) being arbitrary, the integral formula (3.3) can be rearranged as for z ∈ C \ [0, ∞) and arg z ∈ (0, 2π). Let Then f (z) = F (z − b). Therefore, from (3.4), it follows that 2π). The integral representation (3.1) is thus proved. Differentiating n ≥ 0 times with respect to z on both sides of (3.1) and taking the limit z → 0 yield As a result, by virtue of (1.11), we have The integral representation (3.2) for generalized Catalan numbers of the second kind C n (a, b) is thus proved. The proof of Theorem 3.1 is complete.

Potential directions to further study
In this section, we will try to point out two potential directions to further study.

4.1.
Generalized Catalan function of the second kind. Motivated by the integral representation (3.2) for generalized Catalan numbers of the second kind C n (a, b), we can consider the function and call it generalized Catalan function of the second kind, while calling C(a, b; z) in (1.3) generalized Catalan function of the first kind. We can study generalized Catalan function of the second kind C(a, b; z) as a function of three variables a, b, and z. It is easy to see that where the rising factorial (z) n is defined [36,41] by which is also known as the Pochhammer symbol or shifted factorial in the theory of special functions [17,54]. This means that generalized Catalan function of the second kind C(a, b; z) is a completely monotonic function [23,49,56] with respect to b ∈ (0, ∞). Utilizing complete monotonicity [20,49,56], we can derive many new analytic properties of generalized Catalan function of the second kind C(a, b; z). In a word, employing the integral representation (4.1), we believe that we can discover some new properties of generalized Catalan function of the second kind C(a, b; z), of generalized Catalan numbers of the second kind C n (a, b), and the Catalan numbers C n . For the sake of length limit of this paper, we would not like to further study in details.  For more information on results at this point, please refer to [1,2,4,7,9,11,16,26,27,45,52,57] and closely related references therein. Combining (4.2) with (1.11) and (3.2) arrives at

Central binomial coefficients. It is known that
The last three integral representations should provide effective tools for further studying central binomial coefficients 2n n . There are several extensions of central binomial coefficients 2n n in the paper [55]. Remark 4.1. This paper is a revised version of the preprint [33] and a companion of the electronic preprint [12,13].

Declarations
Acknowledgements: The authors thank anonymous referees for their careful corrections to, valuable comments on, and helpful suggestions to the original version of this paper. Funding: Not applicable. Availability of data and material: Data sharing is not applicable to this article as no new data were created or analyzed in this study. Competing interests: The authors declare that they have no conflict of competing interests. Authors' contributions: All authors contributed equally to the manuscript and read and approved the final manuscript.