A Uniﬁed Generalization of the Catalan, Fuss, and Fuss—Catalan Numbers

: In the paper, the authors introduce a uniﬁed generalization of the Catalan numbers, the Fuss numbers, the Fuss–Catalan numbers


Introduction
As well known from [1,2], Catalan numbers C n are used in the study of set partitions in different areas of mathematics.In particular, in combinatorial mathematics, the Catalan numbers C n form a sequence of natural numbers that occur in various counting problems, often involving recursively-defined objects.There are many counting problems in combinatorics whose solution is given by the Catalan numbers C n .The book [3] contains a set of exercises which describe 66 different interpretations of the Catalan numbers.
The Catalan numbers C n can be generated by One of the explicit formulas of C n for n ≥ 0 reads that where Γ(z) = ∞ 0 t z−1 e −t d t, (z) > 0 is the classical Euler gamma function.In [2,4,5] and ( [1] pp. 110-111), it was mentioned that there exists an asymptotic expansion for the Catalan function C x .For new developments on (1), see [6] and the review paper in [7], in which there are plenty of closely related references.
A generalization of the Catalan numbers C n was defined in [8,9] by for n, p ≥ 1.It is obvious that C n = 2 d n .In ( [1] pp. 375-376), the generalization p+1 d n of the Catalan numbers C n is denoted by C(n, p) for p ≥ 0 and is called as the generalized Catalan numbers.In ( [1] pp. 377-378), the Fuss numbers were given and discussed.It is apparent that F(2, n) = C n .
In combinatorial mathematics and statistics, the Fuss-Catalan numbers A n (p, r) are defined [10] as numbers of the form It is easy to see that A n (p, 1) = F(p, n), A n (2, 1) = C n , n ≥ 0 and A n−1 (p, p) = p d n = C(n, p − 1), n, p ≥ 1.
In [22], an alternative and analytical generalization of the Catalan numbers C n and the Catalan function C x was introduced by For uniqueness and convenience of referring to the quantity C(a, b; z), we call the quantity C(a, b; z) the Catalan-Qi function and, when taking z = n ≥ 0, call C(a, b; n) the Catalan-Qi numbers.It is not difficult to verify that C 1 2 , 2; n = C n and for m, n ≥ 1.In [22], it was obtained that for (a), (b) > 0 and (z) ≥ 0. Recently, we discovered in ( [23] Theorem 1.1) relations between the Fuss-Catalan numbers A n (p, r) and the Catalan-Qi numbers C(a, b; n), one of which reads that for integers n ≥ 0, p > 1, and r > 0. In recent papers [6,[22][23][24][25][26][27][28][29][30][31][32], among other things, some properties, including the general expression and a generalization of the asymptotic expansion ( In this paper, we will introduce a unified generalization of the Catalan numbers C n , the generalized Catalan numbers C(n, k), the Fuss numbers F(k, n), the Fuss-Catalan numbers A n (p, r), and the Catalan-Qi function C(a, b; z).Hereafter, we will find a product-ratio expression, similar to the product-ratio expression (4), of the unified generalization in terms of the Catalan-Qi function C(a, b; z).Furthermore, based on the integral representation (3), on the Gauss multiplication formula for the gamma function, and on an integral representation for the logarithm of the gamma function Γ(z), we will derive three integral representations of the unified generalization.Finally, we will establish the logarithmically complete monotonicity of the second order for the unified generalization.

A Unified Generalization of the Catalan and Other Numbers
Does there exist a unified and analytic generalization of the Catalan numbers C n , the Fuss numbers F(m, n) = C(n, m), the Fuss-Catalan numbers A n (p, r), and the Catalan-Qi function C(a, b; z)?What is it concretely?In the early morning of September 15th 2015, a unified generalization was framed out eventually and successfully, and which can be described by a five-variable function where (a), (b) > 0, (p), (q) > 0, and (z) ≥ 0. For uniqueness and convenience of referring to the quantity Q(a, b; p, q; z), we call Q(a, b; p, q; z) the Fuss-Catalan-Qi function and, when taking z = n ≥ 0, call Q(a, b; p, q; n) the Fuss-Catalan-Qi numbers.
It is easy to see that Accordingly, the Fuss-Catalan-Qi function Q(a, b; p, q; z) is a unified generalization of the Catalan numbers C n , the generalized Catalan numbers C(n, m), the Fuss numbers F(m, n), the Fuss-Catalan numbers A n (p, r), and the Catalan-Qi function C(a, b; z).
It is easy to see that the Fuss-Catalan-Qi function Q(a, b; p, q; z) meets Q(b, a; p, q; z) = a b 2(q−p)z 1 Q(a, b; q, p; z) and, when p = q or a = b, Q(a, b; q, p; z)Q(b, a; p, q; z) = 1.
If only swapping p and q, then Γ(rz + b) .

A Product-Ratio Expression of the Fuss-Catalan-Qi Function
Motivated by the product-ratio expression (4), we now find out a product-ratio expression of the Fuss-Catalan-Qi function Q(a, b; p, q; z).Theorem 1.For (a), (b) > 0 and (z) ≥ 0, when p, q ∈ N, we have Proof.By the Gauss multiplication formula in ([33] p. 256, 6.1.20),the Fuss-Catalan-Qi function Q(a, b; p, q; z) can be written as The identity ( 7) is thus proved.The proof of Theorem 1 is complete.
Remark 1.Before getting (7), we did not appreciate the analytic meanings of the form of the product-ratio expression (4) because before catching sight of the unified generalization (5), we did not appreciate the analytic meanings of the form of the Fuss-Catalan numbers A n (p, r) in (2).
Substituting this into (5) and simplifying yields the integral representation in Theorem 2. The proof of Theorem 2 is complete.
By the Gauss multiplication formula (8) and an integral representation for the logarithm of the gamma function Γ(z), we can acquire the second integral representation of the Fuss-Catalan-Qi function Q(a, b; p, q; z), which is seemingly simpler than the one in Theorem 2.
Only by the integral representation (11), we can establish the third integral representation of the Fuss-Catalan-Qi function Q(a, b; p, q; z), which is seemingly simpler than the previous ones.
Proof.By virtue of ( 11), we obtain Substituting this into (5) leads to the integral representation (12).The proof of Theorem 4 is complete.
Remark 4. From (6) and the integral representation in Theorem 2, we obtain Remark 5. When p = q, the integral representation in Theorem 2 is reduced to 1 − e −t/q e −at/q − e −bt/q d t .
Remark 6.By the integral representation (12) and the second relation in (6), we find Remark 7. The function Q(a, b; p, q; z) defined by (5) can be rewritten as Taking the logarithm of G(c, r, z) and differentiating gives Therefore, we obtain for k ∈ {0} ∪ N. Further making use of (15) arrives at q − p)e −(z+1)t − q k+1 e −(qz+b)t + p k+1 e −(pz+a)t d t for k ∈ N. Consequently, for k ∈ N, we have q − p)e −(z+1)t − q k+1 e −(qz+b)t + p k+1 e −(pz+a)t d t.

Properties of the Fuss-Catalan-Qi Function
Recall from ( [35] Chapter XIII), ( [36] Chapter 1), and ( [37] Chapter IV) that an infinitely differentiable function f is said to be completely monotonic on an interval I if it satisfies 0 ≤ (−1) k f (k) (x) < ∞ on I for all k ≥ 0.
Recall from [38,39] that an infinitely differentiable and positive function f is said to be logarithmically completely monotonic on an interval I if 0 ≤ (−1) k [ln f (x)] (k) < ∞ holds on I for all k ∈ N.For more information on logarithmically completely monotonic functions, please refer to [40][41][42][43] and plenty of references therein.
Recall from [38] that if f (k) (x) for some nonnegative integer k is completely monotonic on an interval I but f (k−1) (x) is not completely monotonic on I, then f (x) is called a completely monotonic function of the k-th order on an interval I.
Stimulated by the above definitions and main results in [44], we now introduce the concept of logarithmically completely monotonic functions of the k-th order.Definition 1.For a positive function f (x) on an interval I, if [ln f (x)] (k) for some nonnegative integer k is completely monotonic on an interval I but [ln f (x)] (k−1) is not completely monotonic on I, then we call f (x) a logarithmically completely monotonic function of the k-th order on I.
In terms of the terminology of logarithmically completely monotonic functions of the k-th order, we can state the main results of this section as the following theorem.1. if a < b and q ≤ ln b−ln a ψ(b)−ψ(a) , the function Q(a, b; q, q; x) is increasing on [0, ∞); 2. if a > b and q ≤ ln b−ln a ψ(b)−ψ(a) , the function Q(a, b; q, q; x) is decreasing on [0, ∞); 3. if a < b and q > ln b−ln a ψ(b)−ψ(a) , the function Q(a, b; q, q; x) has a unique minimum on (0, ∞); 4. if a > b and q > ln b−ln a ψ(b)−ψ(a) , the function Q(a, b; q, q; x) has a unique maximum on (0, ∞); 5. if and only if a ≶ b, the function [Q(a, b; q, q; x)] ± is logarithmically completely monotonic of the second order on [0, ∞); in particular, if and only if a ≶ b, the function [Q(a, b; q, q; x)] ±1 is logarithmically convex on [0, ∞).
Proof.Taking the logarithm on both sides of Equation ( 5) and differentiating with respect to x yields and where the asymptotic expansion as z → ∞ in | arg z| < π (see [33] p. 259, 6.3.18) was used, and B k stands for the Bernoulli numbers that are defined by When p = q, making use of in ([33] p. 260, 6.4.1) leads to which means that the derivative ± d 2 [ln Q(a,b;q,q;x)] Hence, the first derivative ± d[ln Q(a,b;q,q;x)] d x is increasing on [0, ∞) if and only if a ≶ b.Meanwhile, the limits ( 13) and ( 14 has a unique zero, which is a maximum point of ln Q(a, b; q, q; x), on (0, ∞).
Therefore, the conclusions on Q(a, b; q, q; x) are thus proved.The proof of Theorem 5 is complete.

Remarks
Finally, we list additional several remarks.[22], we had better leave the combinatorial interpretation of the Fuss-Catalan-Qi function Q(a, b; p, q; z) to combinatorialists and number theorists.
Remark 11.This paper is a corrected and revised version of the preprint [54].

Conclusions
In this paper, we introduce a unified generalization of the Catalan numbers, the Fuss numbers, the Fuss-Catalan numbers, and the Catalan-Qi function, and discover some properties of the unified generalization, including a product-ratio expression of the unified generalization in terms of the Catalan-Qi functions, three integral representations of the unified generalization, and the logarithmically complete monotonicity of the second order for a special case of the unified generalization.

Remark 8 .
Combining Theorem 5 and the last relation in(6), we obtain that the Catalan-Qi function C(a, b; x) is a logarithmically completely monotonic function of the second order.

Remark 9 .
Similar to the introduction of the Catalan-Qi function C(a, b; z) in 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 numbers C(a, b; n), the Catalan-Qi function C(a, b; z), and the Fuss-Catalan numbers A n (p, r) were established.