Some Properties of the Fuss–Catalan Numbers

Abstract

In the paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan numbers, and derive a double inequality for bounding the Fuss–Catalan numbers.

1. Introduction and Main Results

The Catalan numbers $C_n$ for $n\ge 0$ constitute a sequence that is one of the most fascinating sequences in combinatorial number theory with over fifty significant combinatorial interpretations. For details, please refer to monographs [1,2] and closely related references therein.

The Catalan numbers $C_n$ have a generating function

$$\frac{2}{1+\sqrt{1-4x}}=\frac{1-\sqrt{1-4x}}{2x}=\sum _{n=0}^{\infty }{C}_n{x}^n=1+x+2x^2+5x^3+14x^4+42x^5+132x^6+\cdots .$$

Two explicit formulas for $C_n$ with $n\ge 0$ read that

$$C_n=\frac{\left(2n\right)!}{n!(n+1)!}=\frac{4^n\mathsf{\Gamma }(n+1/2)}{\sqrt{\pi }\mathsf{\Gamma }(n+2)},$$

(1)

where

$$\mathsf{\Gamma }\left(z\right)={\int }_0^{\infty }{t}^{z-1}{e}^{-t}dt,\mathsf{\Re }\left(z\right)>0$$

is the classical Euler gamma function. In [1,3,4], it was mentioned that there exists an asymptotic expansion

$$C_x=\frac{4^x\mathsf{\Gamma }(x+1/2)}{\sqrt{\pi }\mathsf{\Gamma }(x+2)}\sim \frac{4^x}{\sqrt{\pi }}\left(\frac{1}{x^{3/2}}-\frac{9}{8}\frac{1}{x^{5/2}}+\frac{145}{128}\frac{1}{x^{7/2}}+\cdots \right),x\to \infty .$$

(2)

In the newly published papers [5,6,7,8,9,10], there are some new results on the Catalan numbers $C_n$ and others.

In [11], an alternative and analytical generalization of the Catalan numbers $C_n$ and the Catalan function $C_x$ was introduced as

$$C(a,b;z)=\frac{\mathsf{\Gamma }\left(b\right)}{\mathsf{\Gamma }\left(a\right)}{\left(\frac{b}{a}\right)}^z\frac{\mathsf{\Gamma }(z+a)}{\mathsf{\Gamma }(z+b)},\mathsf{\Re }\left(a\right),\mathsf{\Re }\left(b\right)>0,\mathsf{\Re }\left(z\right)\ge 0.$$

(3)

For uniqueness and convenience of referring to the quantity $C(a,b;x)$, we call $C(a,b;x)$ the Catalan–Qi function and, when taking $x=n\ge 0$, call $C(a,b;n)$ the Catalan–Qi numbers. Comparing with the second formula in (1) and the first equality in (2), it is clear that

$$C\left(\frac{1}{2},2;x\right)=C_x,x\ge 0.$$

(4)

By the definition (3), we easily see that

$$C(a,b;x)C(b,c;x)=C(a,c;x),a,b,c>0,x\ge 0.$$

In the papers [11,12,13,14,15,16,17,18,19,20,21,22,23], the authors discovered many analytic properties, including the monotonicity, a general expression of the asymptotic expansion (2), Schur-convexity, a generalization of the expansion (2), minimality, (logarithmically) complete monotonicity, product inequalities, a generating function, logarithmic convexity, exponential representations, determinantal inequalities, series identities, integral representations, and connections with the Bessel polynomials and the Bell polynomials of the second kind, of the Catalan numbers and function $C_n$ and $C_x$ and the Catalan–Qi function $C(a,b;x)$.

In combinatorial mathematics and statistics, the Fuss–Catalan numbers $A_n(p,r)$ are defined in [24] as numbers of the form

$$A_n(p,r)=\frac{r}{np+r}\left(\genfrac{}{}{0pt}{}{np+r}{n}\right)=r\frac{\mathsf{\Gamma }(np+r)}{\mathsf{\Gamma }(n+1)\mathsf{\Gamma }\left(n\right(p-1)+r+1)}.$$

(5)

Comparing with the first formula in (1), it is obvious that

$$A_n(2,1)=C_n,n\ge 0.$$

(6)

A generalization of the Catalan numbers $C_n$ was defined in [25,26,27] by

$${}_p{d}_n=\frac{1}{n}\left(\genfrac{}{}{0pt}{}{pn}{n-1}\right)=\frac{1}{(p-1)n+1}\left(\genfrac{}{}{0pt}{}{pn}{n}\right)$$

for $n\ge 1$. The usual Catalan numbers $C_n={}_2{d}_n$ are a special case with $p=2$. It is immediate that

$$A_{n-1}(p,p)={}_p{d}_n$$

for $n\ge 1$. There exists some literature such as [28,29,30,31,32,33,34,35,36,37,38] devoted to the investigation of the Fuss–Catalan numbers $A_n(p,r)$.

Considering the relations (4) and (6), one may ask a question: what is the relation between the Catalan–Qi numbers $C(a,b;n)$ and the Fuss–Catalan numbers $A_n(p,r)$? This question is answered by Theorem 1 below.

Theorem 1.

For $n,r\ge 0$ and $p>1$, we have

$$A_n(p,r)=r^n\frac{{\prod }_{k=1}^{p}C\left(\frac{k+r-1}{p},1;n\right)}{{\prod }_{k=1}^{p-1}C\left(\frac{k+r}{p-1},1;n\right)}.$$

(7)

For $n,p\in \mathbb{N}$ and $r\ge 0$, we have

$$A_n(p,r)=\frac{r}{nB\left(n\right(p-1)+1,n)}\frac{{\left(np\right)}^{r-1}}{{[n(p-1)+1]}^r}C(np,n(p-1)+1;r),$$

(8)

where $B(x,y)$ denotes the classical beta function. For $r+1>n>0$ and $p\ge 0$, we have

$$A_n(p,r)=\frac{1}{nB(n,r-n+1)}{\left(\frac{r}{r-n+1}\right)}^{np}C(r,r-n+1;np).$$

(9)

When $r+1>n\ge 1$ and $p\ge 0$, we have

$$A_n(p,r)=\frac{1}{n}\frac{{\left[B(r+1-n,n)\right]}^{p-1}}{{\left[B(r,n)\right]}^p}\prod _{k=0}^{n-1}C\left(\frac{r+k}{n},\frac{r-n+k+1}{n};p\right).$$

(10)

For $n\ge 2$, $r+1>n$, and $p\in \mathbb{N}$, we have

$$A_n(p,r)=r{p}^{1/2}B(n-1,2)\frac{{\left[B(r+1-n,n-1)\right]}^{n-1}}{{\left[B(r+1-n+p,n-1)\right]}^n}\prod _{k=0}^{p-1}C\left(\frac{r+k}{p},\frac{r+k+1-n}{p};n\right).$$

(11)

Recall from ([39], pp. 372–373) and ([40], p. 108, Definition 4) that a sequence ${\left\{{\mu }_n\right\}}_{0\le n\le \infty }$ is said to be completely monotonic if its elements are non-negative and its successive differences alternate sign, that is,

$${(-1)}^k{\Delta }^k{\mu }_n\ge 0,n,k\ge 0,$$

where

$${\Delta }^k{\mu }_n=\sum _{m=0}^k{(-1)}^m\left(\genfrac{}{}{0pt}{}{k}{m}\right){\mu }_{n+k-m}.$$

Further recall from ([40], p. 163, Definition 14a) that a completely monotonic sequence ${\left\{a_n\right\}}_{n\ge 0}$ is minimal if it ceases to be completely monotonic when $a_0$ is decreased.

Applying the identity (7) and several analytic properties of the Catalan–Qi function $C(a,b;x)$, we find several analytic properties, including monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan sequence ${\left\{A_n(p,r)\right\}}_{n\ge 0}$ and related ones.

Theorem 2.

When $p\ge r>0$,

1.

the sequence ${\left\{{\mathcal{A}}_n(p,r)\right\}}_{n\ge 0}$,

$${\mathcal{A}}_n(p,r)=\left\{\begin{array}{cc}1,\hfill & n=0,\hfill \\ {\displaystyle \frac{1}{\sqrt[n]{A_n(p,r)}}},\hfill & n\in \mathbb{N},\hfill \end{array}\right.$$

(12)

is logarithmically convex, completely monotonic, and minimal;

2.

the sequence of the Fuss–Catalan numbers ${\left\{A_n(p,r)\right\}}_{n\ge 0}$ is increasing and logarithmically convex.

Finally, by applying a double inequality of the beta function in the papers [41] and ([42], pp. 78–81, Section 3), we derive a double inequality for the Fuss–Catalan numbers $A_n(p,r)$.

Theorem 3.

For $n\ge 2$ and $p,r\in \mathbb{N}$, we have

$$A_n(p,r)\ge \frac{r\left[n\right(p-1)+r+1]}{n}.$$

(13)

When $m\triangleq min\{D(n-1)D(n(p-1)+r+1),b_A(n-1)[n(p-1)+r+1]\}<1$ for $n\ge 2$ and $p,r\in \mathbb{N}$, we have

$$A_n(p,r)\le \frac{r\left[n\right(p-1)+r+1]}{n(1-m)},$$

where

$$D\left(x\right)=\frac{x-1}{\sqrt{2x-1}}\mathrm{and}{b}_A=\underset{x\ge 1}{max}\left[\frac{1}{x^2}-\frac{{\mathsf{\Gamma }}^2\left(x\right)}{\mathsf{\Gamma }\left(2x\right)}\right]=0.08731\cdots$$

(14)

for $x\ge 1$.

2. Lemmas

In order to prove Theorems 2 and 3, we need the following notion and lemmas.

Recall from [43,44,45] that an infinitely differentiable and positive function f is said to be logarithmically completely monotonic on an interval I if $0\le {(-1)}^k{[lnf\left(x\right)]}^{\left(k\right)}<\infty$ holds on I for all $k\in \mathbb{N}$.

Lemma 1

([18], Theorem 6). The function

$${\mathcal{C}}^{\pm 1}(a,b;x)=\left\{\begin{array}{cc}1,\hfill & x=0\hfill \\ {\left[C\right(a,b;x\left)\right]}^{\pm 1/x},\hfill & x>0\hfill \end{array}\right.$$

is logarithmically completely monotonic on $[0,\infty )$ if and only if $a\gtrless b$.

Lemma 2

([18], Theorem 7). Let $a,b>0$ and $x\ge 0$. Then

1.

the unique zero $x_0$ of the equation

$$\frac{\psi (x+b)-\psi (x+a)}{lnb-lna}=1$$

satisfies $x_0\in \left(0,\frac{1}{2}\right)$, where ψ is the logarithmic derivative of the gamma function Γ;

2.

when $b>a$, the function $C(a,b;x)$ is decreasing in $x\in [0,x_0)$, increasing in $x\in (x_0,\infty )$, and logarithmically convex in $x\in [0,\infty )$;

3.

when $b<a$, the function $C(a,b;x)$ is increasing in $x\in [0,x_0)$, decreasing in $x\in (x_0,\infty )$, and logarithmically concave in $x\in [0,\infty )$.

Lemma 3

([41] and [42] pp. 78–81, Section 3). For $x,y>1$, we have

$$0\le \frac{1}{xy}-B(x,y)\le min\left\{\frac{D\left(x\right)D\left(y\right)}{xy},b_A\right\},$$

(15)

where $D\left(x\right)$ and $b_A$ are defined as in (14).

3. Proofs of Theorems 1–3

We now start out to prove our theorems.

Proof of Theorem 1.

By virtue of the Gauss multiplication formula

$$\mathsf{\Gamma }\left(nz\right)=\frac{n^{nz-1/2}}{{\left(2\pi \right)}^{(n-1)/2}}\prod _{k=0}^{n-1}\mathsf{\Gamma }\left(z+\frac{k}{n}\right)$$

(16)

from ([46], p. 256, 6.1.20), the Fuss–Catalan numbers $A_n(p,r)$ defined by the second expression in (5) can also be rewritten as

$$\begin{array}{c}{A}_n(p,r)=\frac{r\mathsf{\Gamma }\left(p\right(n+r/p\left)\right)}{\mathsf{\Gamma }(n+1)\mathsf{\Gamma }\left(\right(p-1\left)\right(n+(r+1)/(p-1)\left)\right)}\hfill \\ =\frac{r\frac{p^{np+r-1/2}}{{\left(2\pi \right)}^{(p-1)/2}}{\prod }_{k=0}^{p-1}\mathsf{\Gamma }\left(n+\frac{k+r}{p}\right)}{\mathsf{\Gamma }(n+1)\frac{{(p-1)}^{n(p-1)+r+1/2}}{{\left(2\pi \right)}^{(p-2)/2}}{\prod }_{k=0}^{p-2}\mathsf{\Gamma }\left(n+\frac{k+r+1}{p-1}\right)}\hfill \\ =\frac{1}{\sqrt{2\pi }}{\left(\frac{p}{p-1}\right)}^r\frac{r}{{\left[p(p-1)\right]}^{1/2}}{\left[\frac{p^p}{{(p-1)}^{p-1}}\right]}^n\frac{\mathsf{\Gamma }\left(1\right)}{\mathsf{\Gamma }\left(\frac{r}{p}\right)}{\left(\frac{1}{r/p}\right)}^n\frac{\mathsf{\Gamma }\left(n+\frac{r}{p}\right)}{\mathsf{\Gamma }(n+1)}\hfill \\ \times \prod _{k=1}^{p-1}\frac{\mathsf{\Gamma }\left(\frac{k+r}{p-1}\right)}{\mathsf{\Gamma }\left(\frac{k+r}{p}\right)}{\left(\frac{\frac{k+r}{p-1}}{\frac{k+r}{p}}\right)}^n\frac{\mathsf{\Gamma }\left(n+\frac{k+r}{p}\right)}{\mathsf{\Gamma }\left(n+\frac{k+r}{p-1}\right)}\frac{\mathsf{\Gamma }\left(\frac{r}{p}\right)}{\mathsf{\Gamma }\left(1\right)}\prod _{k=1}^{p-1}\frac{\mathsf{\Gamma }\left(\frac{k+r}{p}\right)}{\mathsf{\Gamma }\left(\frac{k+r}{p-1}\right)}{\left(\frac{r/p}{1}\right)}^n\prod _{k=1}^{p-1}{\left(\frac{\frac{k+r}{p}}{\frac{k+r}{p-1}}\right)}^n\hfill \\ =\frac{1}{\sqrt{2\pi }}{\left(\frac{p}{p-1}\right)}^r\frac{r^{n+1}}{{\left[p(p-1)\right]}^{1/2}}\frac{\mathsf{\Gamma }\left(\frac{r}{p}\right)}{\mathsf{\Gamma }\left(1\right)}\prod _{k=1}^{p-1}\frac{\mathsf{\Gamma }\left(\frac{k+r}{p}\right)}{\mathsf{\Gamma }\left(\frac{k+r}{p-1}\right)}C\left(\frac{r}{p},1;n\right)\prod _{k=1}^{p-1}C\left(\frac{k+r}{p},\frac{k+r}{p-1};n\right)\hfill \\ =\frac{1}{\sqrt{2\pi }}{\left(\frac{p}{p-1}\right)}^r\frac{1}{{\left[p(p-1)\right]}^{1/2}}\frac{{\prod }_{k=0}^{p-1}\mathsf{\Gamma }\left(\frac{r}{p}+\frac{k}{p}\right)}{{\prod }_{k=0}^{p-2}\mathsf{\Gamma }\left(\frac{r+1}{p-1}+\frac{k}{p-1}\right)}{r}^{n+1}C\left(\frac{r}{p},1;n\right)\prod _{k=1}^{p-1}C\left(\frac{k+r}{p},\frac{k+r}{p-1};n\right)\hfill \\ =r^{n}C\left(\frac{r}{p},1;n\right)\prod _{k=1}^{p-1}C\left(\frac{k+r}{p},\frac{k+r}{p-1};n\right).\hfill \end{array}$$

Further making use of

$$C(a,b;x)=\frac{1}{C(b,a;x)},a,b>0,x\ge 0,$$

We can rearrange the above result as

$$\begin{array}{c}{A}_n(p,r)=r^{n}C\left(\frac{r}{p},1;n\right)\prod _{k=1}^{p-1}\left[C\left(\frac{k+r}{p},1;n\right)C\left(1,\frac{k+r}{p-1};n\right)\right]=r^{n}C\left(\frac{r}{p},1;n\right)\prod _{k=1}^{p-1}\frac{C\left(\frac{k+r}{p},1;n\right)}{C\left(\frac{k+r}{p-1},1;n\right)}\\ =r^{n}C\left(\frac{r}{p},1;n\right)\frac{{\prod }_{k=1}^{p-1}C\left(\frac{k+r}{p},1;n\right)}{{\prod }_{k=1}^{p-1}C\left(\frac{k+r}{p-1},1;n\right)}=r^n\frac{{\prod }_{k=0}^{p-1}C\left(\frac{k+r}{p},1;n\right)}{{\prod }_{k=1}^{p-1}C\left(\frac{k+r}{p-1},1;n\right)}=r^n\frac{{\prod }_{k=1}^{p}C\left(\frac{k+r-1}{p},1;n\right)}{{\prod }_{k=1}^{p-1}C\left(\frac{k+r}{p-1},1;n\right)}.\end{array}$$

The identity (7) thus follows.

It is straightforward that

$$\begin{array}{cc}\hfill A_n(p,r)& =\frac{r}{\mathsf{\Gamma }(n+1)}\frac{\mathsf{\Gamma }(r+np)}{\mathsf{\Gamma }(r+np-n+1)}\hfill \\ & =\frac{r}{\mathsf{\Gamma }(n+1)}\frac{\mathsf{\Gamma }\left(np\right)}{\mathsf{\Gamma }(np-n+1)}{\left(\frac{np}{np-n+1}\right)}^{r}C(np,np-n+1;r)\hfill \\ & =\frac{r\mathsf{\Gamma }\left(n\right)}{np\mathsf{\Gamma }(n+1)}\frac{\mathsf{\Gamma }(np+1)}{\mathsf{\Gamma }(np-n+1)\mathsf{\Gamma }\left(n\right)}{\left(\frac{np}{np-n+1}\right)}^{r}C(np,np-n+1;r)\hfill \\ & =\frac{r}{nB(np-n+1,n)}\frac{{\left(np\right)}^{r-1}}{{(np-n+1)}^r}C(np,np-n+1;r)\hfill \end{array}$$

and, interchanging the role of r and $np$,

$$\begin{array}{cc}\hfill A_n(p,r)& =\frac{r}{\mathsf{\Gamma }(n+1)}\frac{\mathsf{\Gamma }(np+r)}{\mathsf{\Gamma }(np+r-n+1)}\hfill \\ & =\frac{r}{\mathsf{\Gamma }(n+1)}\frac{\mathsf{\Gamma }\left(r\right)}{\mathsf{\Gamma }(r-n+1)}{\left(\frac{r}{r-n+1}\right)}^{np}C(r,r-n+1;np)\hfill \\ & =\frac{1}{n}\frac{\mathsf{\Gamma }(r+1)}{\mathsf{\Gamma }\left(n\right)\mathsf{\Gamma }(r-n+1)}{\left(\frac{r}{r-n+1}\right)}^{np}C(r,r-n+1;np)\hfill \\ & =\frac{1}{nB(n,r-n+1)}{\left(\frac{r}{r-n+1}\right)}^{np}C(r,r-n+1;np).\hfill \end{array}$$

Therefore, the identities (8) and (9) follow.

By the Gauss multiplication formula (16) again, when $r+1>n$, the Fuss–Catalan numbers $A_n(p,r)$ defined by the second expression in (5) can be rearranged as

$$\begin{array}{c}{A}_n(p,r)=\frac{r}{\mathsf{\Gamma }(n+1)}\frac{\frac{n^{np+r-1/2}}{{\left(2\pi \right)}^{(n-1)/2}}{\prod }_{k=0}^{n-1}\mathsf{\Gamma }\left(p+\frac{r}{n}+\frac{k}{n}\right)}{\frac{n^{n(p-1)+r+1/2}}{{\left(2\pi \right)}^{(n-1)/2}}{\prod }_{k=0}^{n-1}\mathsf{\Gamma }\left(p+\frac{r-n+1}{n}+\frac{k}{n}\right)}\hfill \\ \hfill \begin{array}{c}=\frac{n^{n-1}r}{\mathsf{\Gamma }(n+1)}\prod _{k=0}^{n-1}\frac{\mathsf{\Gamma }\left(\frac{r+k}{n}\right)}{\mathsf{\Gamma }\left(\frac{r-n+k+1}{n}\right)}{\left(\frac{r+k}{r-n+k+1}\right)}^p\prod _{k=0}^{n-1}\frac{\mathsf{\Gamma }\left(\frac{r-n+k+1}{n}\right)}{\mathsf{\Gamma }\left(\frac{r+k}{n}\right)}{\left(\frac{r-n+k+1}{r+k}\right)}^p\frac{\mathsf{\Gamma }\left(p+\frac{r+k}{n}\right)}{\mathsf{\Gamma }\left(p+\frac{r-n+k+1}{n}\right)}\hfill \\ =\frac{n^{n-1}r}{\mathsf{\Gamma }(n+1)}\frac{\frac{{\left(2\pi \right)}^{(n-1)/2}}{n^{r-1/2}}\mathsf{\Gamma }\left(r\right)}{\frac{{\left(2\pi \right)}^{(n-1)/2}}{n^{r-n+1/2}}\mathsf{\Gamma }(r-n+1)}{\left[\frac{\mathsf{\Gamma }(r+n)\mathsf{\Gamma }(r+1-n)}{\mathsf{\Gamma }\left(r\right)\mathsf{\Gamma }(r+1)}\right]}^p\prod _{k=0}^{n-1}C\left(\frac{r+k}{n},\frac{r-n+k+1}{n};p\right)\hfill \\ =\frac{\mathsf{\Gamma }\left(n\right)}{\mathsf{\Gamma }(n+1)}{\left[\frac{\mathsf{\Gamma }(r+n)}{\mathsf{\Gamma }\left(r\right)\mathsf{\Gamma }\left(n\right)}\right]}^p{\left[\frac{\mathsf{\Gamma }(r+1-n)\mathsf{\Gamma }\left(n\right)}{\mathsf{\Gamma }(r+1)}\right]}^{p-1}\prod _{k=0}^{n-1}C\left(\frac{r+k}{n},\frac{r-n+k+1}{n};p\right)\hfill \\ =\frac{1}{n}\frac{{\left[B(r+1-n,n)\right]}^{p-1}}{{\left[B(r,n)\right]}^p}\prod _{k=0}^{n-1}C\left(\frac{r+k}{n},\frac{r-n+k+1}{n};p\right).\hfill \end{array}\end{array}$$

The identity (10) is thus proved.

Similarly, by virtue of (16) once again, we have

$$\begin{array}{c}{A}_n(p,r)=\frac{r{p}^{n-1}}{\mathsf{\Gamma }(n+1)}\prod _{k=0}^{p-1}\frac{\mathsf{\Gamma }\left(n+\frac{r+k}{p}\right)}{\mathsf{\Gamma }\left(n+\frac{r+k+1-n}{p}\right)}\hfill \\ \hfill \begin{array}{c}=\frac{r{p}^{n-1}}{\mathsf{\Gamma }(n+1)}\prod _{k=0}^{p-1}\frac{\mathsf{\Gamma }\left(\frac{r}{p}+\frac{k}{p}\right)}{\mathsf{\Gamma }\left(\frac{r+1-n}{p}+\frac{k}{p}\right)}{\left(\frac{r+k}{r+k+1-n}\right)}^n\prod _{k=0}^{p-1}C\left(\frac{r+k}{p},\frac{r+k+1-n}{p};n\right)\hfill \\ =\frac{r{p}^{n-1}}{\mathsf{\Gamma }(n+1)}\frac{{\prod }_{k=0}^{p-1}\mathsf{\Gamma }\left(\frac{r}{p}+\frac{k}{p}\right)}{{\prod }_{k=0}^{p-1}\mathsf{\Gamma }\left(\frac{r+1-n}{p}+\frac{k}{p}\right)}{\left(\prod _{k=0}^{p-1}\frac{r+k}{r+k+1-n}\right)}^n\prod _{k=0}^{p-1}C\left(\frac{r+k}{p},\frac{r+k+1-n}{p};n\right)\hfill \\ =\frac{r{p}^{n-1}}{\mathsf{\Gamma }(n+1)}\frac{p^{r+1-n}\mathsf{\Gamma }\left(r\right)}{p^{r-1/2}\mathsf{\Gamma }(r+1-n)}{\left[\frac{\mathsf{\Gamma }(r+p)}{\mathsf{\Gamma }\left(r\right)}\frac{\mathsf{\Gamma }(r+1-n)}{\mathsf{\Gamma }(r+1-n+p)}\right]}^n\prod _{k=0}^{p-1}C\left(\frac{r+k}{p},\frac{r+k+1-n}{p};n\right)\hfill \\ =r{p}^{1/2}B(n-1,2)\frac{{\left[B(r+1-n,n-1)\right]}^{n-1}}{{\left[B(r+1-n+p,n-1)\right]}^n}\prod _{k=0}^{p-1}C\left(\frac{r+k}{p},\frac{r+k+1-n}{p};n\right).\hfill \end{array}\end{array}$$

The identity (11) is demonstrated. The proof of Theorem 1 is complete. ☐

Proof of Theorem 2.

We call (see related chapters [39] Chapter XIII, [45] Chapter 1, and [40] Chapter IV) a positive function f defined on an interval I completely monotonic if all of its derivatives satisfy $0\le {(-1)}^k{f}^{\left(k\right)}\left(x\right)<\infty$ for all $k\ge 0$ on I. The inclusions

$$\mathcal{S}\setminus \left\{0\right\}\subset \mathcal{L}\left[\right(0,\infty \left)\right]\mathrm{and}\mathcal{L}\left[I\right]\subset \mathcal{C}\left[I\right]$$

(17)

were found in [44,47,48], where $\mathcal{S}$, $\mathcal{L}\left[I\right]$, and $\mathcal{C}\left[I\right]$ represent the class of Stieltjes transforms on $(0,\infty )$, the class of logarithmically completely monotonic functions I, and the class of completely monotonic functions on I, respectively. This was mentioned in the monograph [45] and admitted in mathematical community.

By Lemma 1, since $\frac{r}{p}<1$ and $\frac{k+r}{p}<\frac{k+r}{p-1}$ for all $p>1$, $p>r>0$, and $1\le k\le p-1$, the functions

$$\frac{1}{\mathcal{C}(r/p,1;x)}\mathrm{and}\frac{1}{\mathcal{C}\left(\right(k+r)/p,(k+r)/(p-1);x)}$$

are logarithmically completely monotonic on $[0,\infty )$. It is easy to see that the product of finitely many logarithmically completely monotonic functions is still logarithmically completely monotonic. Hence, the function

$$\frac{1}{r\mathcal{C}(r/p,1;x){\prod }_{k=1}^{p-1}\mathcal{C}((k+r)/p,(k+r)/(p-1);x)}$$

(18)

is logarithmically completely monotonic on the interval $[0,\infty )$. Consequently, by the definition of logarithmically completely monotonic functions, the sequence (12) is decreasing and logarithmically convex.

Further, by virtue of the first inclusion in (17) and the logarithmically complete monotonicity of the function (18), the function (18) is also completely monotonic on $[0,\infty )$. By Theorem 14b in ([40] p. 164), we know that a sequence ${\left\{a_n\right\}}_0^{\infty }$ is minimal completely monotonic if and only if there exists a completely monotonic function $f\left(x\right)$ on $[0,\infty )$ such that $f\left(n\right)=a_n$ for $n\ge 0$. Hence, we arrive at the complete monotonicity and minimality of the sequence ${\mathcal{A}}_n(p,r)$ defined by (12).

Similarly, by Lemma 2, the identity (7), and $A_0(p,r)=1$ for all $p>1$ and $r>0$, we conclude that the sequence of the Fuss–Catalan numbers ${\left\{A_n(p,r)\right\}}_{n\ge 0}$ is increasing and logarithmically convex. The proof of Theorem 2 is complete. ☐

Proof of Theorem 3.

Applying the inequality (15) to

$$A_n(p,r)=\frac{r}{n(n-1)}\frac{1}{B(n-1,n(p-1)+r+1)}$$

results in

$$\frac{r\left[n\right(p-1)+r+1]}{n}\le A_n(p,r)$$

and, when $min\{D(n-1)D(n(p-1)+r+1),b_A(n-1)[n(p-1)+r+1]\}<1$,

$$A_n(p,r)\le \frac{r\left[n\right(p-1)+r+1]}{n}\frac{1}{1-min\{D(n-1)D(n(p-1)+r+1),b_A(n-1)[n(p-1)+r+1]\}}$$

which can be rearranged as (13). The proof of Theorem 3 is complete. ☐

4. Remarks

Finally, we give several remarks on our main results.

Remark 1.

The Catalan–Qi function $C(a,b;z)$, the second expression in (5) of the Fuss–Catalan numbers $A_n(p,r)$, and all the identities from (7) to (11) in Theorem 1 can be extended to real values, even to complex values, that the variables $a,b,x$ and $n,p,r$ can take.

Remark 2.

The identity (7) is more beautiful and informative than the others in Theorem 1, because the form of the identity (7) is simpler, more regular, and with less restrictions to n, p, and r.

Remark 3.

Theorem 1 means that the Fuss–Catalan numbers $A_n(p,r)$ can be represented in terms of the Catalan–Qi functions $C(a,b;x)$. This fact shows that introducing the Catalan–Qi function $C(a,b;x)$ in [11] is analytically significant. However, we need the combinatorialists to combinatorially interpret the Catalan–Qi function $C(a,b;x)$ or its special cases.

Remark 4.

By the definition of the classical beta function, we easily see that

$$\frac{\mathsf{\Gamma }(z+a)}{\mathsf{\Gamma }(z+b)}=\frac{B(b-a,z+a)}{\mathsf{\Gamma }(b-a)}.$$

(19)

We can use the inequality (15) and the formula (19) to bound the Catalan numbers $C_n$, the Fuss–Catalan numbers $A_n(p,r)$, and the Catalan–Qi function $C(a,b;x)$.

There has been an amount of literature on the ratio of two gamma functions, see, for example, the expository and survey articles [49,50,51,52,53,54,55] and plenty of references therein. Applying results in these literature, we can estimate the Catalan numbers $C_n$, the Fuss–Catalan numbers $A_n(p,r)$, and the Catalan–Qi function $C(a,b;x)$ in terms of inequalities and asymptotic formulas of the gamma function $\mathsf{\Gamma }\left(x\right)$. For example, the double inequality

$$\begin{array}{c}\sqrt{\frac{b}{a}}{\left[I(a,b)\right]}^{a-b}exp\left[\sum _{j=1}^{2m}\frac{B_{2j}}{2j(2j-1)}\left(\frac{1}{a^{2j-1}}-\frac{1}{b^{2j-1}}\right)\right]<\frac{\mathsf{\Gamma }\left(a\right)}{\mathsf{\Gamma }\left(b\right)}\hfill \\ \hfill <\sqrt{\frac{b}{a}}{\left[I(a,b)\right]}^{a-b}exp\left[\sum _{j=1}^{2m-1}\frac{B_{2j}}{2j(2j-1)}\left(\frac{1}{a^{2j-1}}-\frac{1}{b^{2j-1}}\right)\right]\end{array}$$

for $m\in \mathbb{N}$ and $a,b>0$ was derived in ([18] Theorem 11), where $B_i$ for $i\ge 0$ are the classical Bernoulli numbers generated by

$$\frac{x}{e^x-1}=\sum _{i=0}^{\infty }{B}_i\frac{x^i}{i!}=1-\frac{x}{2}+\sum _{j=1}^{\infty }{B}_{2j}\frac{x^{2j}}{\left(2j\right)!},\left|x\right|<2\pi$$

and

$$I(\alpha ,\beta )=\frac{1}{e}{\left(\frac{{\beta }^{\beta }}{{\alpha }^{\alpha }}\right)}^{1/(\beta -\alpha )}$$

for $\alpha ,\beta >0$ with $\alpha \ne \beta$ stands for the identric mean [56]. One more example is the double inequality

$$e^{\psi \left(L\right(a,b\left)\right)}<{\left[\frac{\mathsf{\Gamma }\left(a\right)}{\mathsf{\Gamma }\left(b\right)}\right]}^{1/(a-b)}<e^{\psi \left(I\right(a,b\left)\right)},a,b>0,a\ne b$$

in [57,58], where $\psi \left(x\right)=\frac{{\mathsf{\Gamma }}^{\prime }\left(x\right)}{\mathsf{\Gamma }\left(x\right)}$ is the digamma function and

$$L(a,b)=\frac{b-a}{lnb-lna},a,b>0,a\ne b$$

is called the logarithmic mean [56].

In the papers [59,60], some inequalities for the beta function $B(x,y)$ are reviewed and surveyed. The main result in ([60], Theorem) states that, for every $k\ge 1$ and every point $x,y>0$, the quantity ${(-1)}^{k-1}{D}_{x,y}^{\left(k\right)}(X,Y)$ is positive for $X,Y>0$ and, if k is even, positive definite in $X,Y$, where $D_{x,y}^{\left(k\right)}(X,Y)$ denotes the kth differential of $F(x+X,y+Y)$ in X and Y and

$$F(x,y)=\frac{\mathsf{\Gamma }(x+1)\mathsf{\Gamma }(y+1)}{\mathsf{\Gamma }(x+y+1)}\frac{{(x+y)}^{x+y}}{x^x{y}^y}.$$

As said in ([60], p. 1430), this theorem can produce lower and upper bounds for $F(x,y)$ and every bound for $F(x,y)$ is actually a bound for $B(x,y)$. By inequalities for the beta function $B(x,y)$, we can derive more inequalities for the Fuss–Catalan numbers $A_n(p,r)$.

To the best of our knowledge, we do not find any known inequality for the Fuss–Catalan numbers $A_n(p,r)$ in published literature. It seems that there are more identities than inequalities in combinatorics.

Remark 5.

In order to give a better combinatorial context, we would like to mention three references [2,61,62]. In [61], a collection of combinatorial interpretations for the Fuss–Catalan numbers $A_n(p;1)$ is presented. In [2], a monograph about the Catalan numbers $C_n$ and their generalizations, the Fuss–Catalan numbers $A_n(p,r)$ are studied in Additional Problem A14. In ([62], Corollary 3.4), there is a conclusion about the sequence ${\left\{A_{n-1}(p,p)\right\}}_{n\ge 0}$.

Remark 6.

This paper is a simplified and corrected version of the preprint [63].

5. Conclusions

In this paper, the authors express the Fuss–Catalan numbers as several forms in terms of the Catalan–Qi function; find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the Fuss–Catalan numbers; and derive a double inequality for bounding the Fuss–Catalan numbers.