Abstract
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.
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Background
It is stated in Koshy (2009), Stanley and Weisstein (2015) that the Catalan numbers \(C_n\) for \(n\ge 0\) 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 \(n\ge 0\) read that
where
is the classical Euler gamma function and
is the generalized hypergeometric series defined for complex numbers \(a_i\in {\mathbb {C}}\) and \(b_i\in {\mathbb {C}}{\setminus} \{0,-1,-2,\ldots \}\), for positive integers \(p,q\in {\mathbb {N}}\), and in terms of the rising factorials \((x)_n\) defined by
and
In Graham et al. (1994), Koshy (2009), Stanley and Weisstein (2015), Vardi (1991), it was mentioned that there exists an asymptotic expansion
for the Catalan function \(C_x\). What is the general expression for the asymptotic expansion (5)?
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 \(x\ge 0\) 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 \(C\bigl (\frac{1}{2},2;x\bigr )\) 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).
In Koshy (2009) and from https://en.wikipedia.org/wiki/Catalan_number, the integral representation
was listed. In Nkwanta and Tefera (2013, p. 10), there is an integral representation
In Qi et al. (2015c, Theorem 1.4), the integral representations
was established. In Qi (2015a, Theorem 1.3), the equivalence relation between (8) and (9) was verified. What is the integral representation of the Catalan–Qi function C(a, b; x) similar to either (8) or (9)?
From the power series (1), we observe that the Catalan numbers \(C_n\) is an increasing sequence in \(n\ge 0\) 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 \(x\ge 0\), 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 \({\mathfrak {R}}(z)>0\) and \({\mathfrak {R}}(w)>0\) 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) in \(x\in [0,\infty )\).
As showed by (1), the Catalan numbers \(C_n\) have a generating function \(\frac{2}{1+\sqrt{1-4x}\,}\). 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.
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).
Lemma 1
Let u(x) and \(v(x)\ne 0\) be differentiable functions, let \(U_{(n+1)\times 1}(x)\) be an \((n+1)\times 1\) matrix whose elements \(u_{k,1}(x)=u^{(k-1)}(x)\) for \(1\le k\le n+1,\) let \(V_{(n+1)\times n}(x)\) be an \((n+1)\times n\) matrix whose elements
for \(1\le i\le n+1\) and \(1\le j\le n,\) and let \(|W_{(n+1)\times (n+1)}(x)|\) denote the determinant of the \((n+1)\times (n+1)\) matrix
Then the nth derivative of the ratio \(\frac{u(x)}{v(x)}\) can be computed by
Making use of the formula (13) in Lemma 1, we can obtain the following new expression for the Catalan numbers \(C_n\).
Theorem 1
For \(n\in {\mathbb {N}}\), the nth derivative of the generating function of the Catalan numbers \(C_n\) can be expressed as
and the Catalan numbers \(C_n\) can be represented as
where \(\langle x\rangle _n\) is the falling factorial defined by
and \((x)_n\) is the rising factorial defined by (3).
Proof
Let \(u(x)=1-\sqrt{1-4x}\) and \(v(x)=x\). Since
for \(k\in {\mathbb {N}}\) as \(x\rightarrow 0\), making use of the formula (13) yields
as \(x\rightarrow 0\). By virtue of the second function in the Eq. (1), we see that
The proof of Theorem 1 is complete. \(\square\)
Asymptotic expansions of the Catalan–Qi function \(\varvec{C(a,b;x)}\)
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 \(\frac{\Gamma (a)}{\Gamma (b)}\) for \(a,b>0\).
Theorem 2
Let \(B_k^{(\sigma )}(x)\) denote the generalized Bernoulli polynomials defined by
For \(b>a>0\), the Catalan–Qi function C(a, b; x) has the asymptotic expansion
as \(x\rightarrow \infty\). Consequently, the Catalan function \(C_x\) has the asymptotic expansion
as \(x\rightarrow \infty\).
Proof
In Temme (1996, p. 67), it was listed that, under the condition \({\mathfrak {R}}(b-a)>0\),
in the sector \(|\arg z|<\pi\), where the generalized Bernoulli polynomials \(B_k^{(\sigma )}(x)\) 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 \(x\rightarrow \infty\). In particular, when taking \(a=\frac{1}{2}\) and \(b=2\) in (15), we obtain the asymptotic expansion (16). Theorem 2 is thus proved. \(\square\)
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 \(a=\frac{1}{2}\) and \(b=2\).
Remark 2
The asymptotic expansion (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 \(I(\alpha ,\beta )\) denotes the exponential mean defined by
for \(\alpha ,\beta >0\) with \(\alpha \ne \beta\). Consequently, we have
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 \(x\rightarrow 0\) in (18) and rearranging. Theorem 3 is thus proved. \(\square\)
Remark 3
When taking \(a=\frac{1}{2}\) and \(b=2\), the asymptotic expansion (18) reduces to one of conclusions in Qi (2015a, Theorem 1.2).
Remark 4
For more information on the exponential mean \(I(\alpha ,\beta )\) in (19), please refer to the monograph (Bullen 2003) and the papers (Guo and Qi 2009, 2011).
Integral representations and complete monotonicity of the Catalan–Qi function \(\varvec{C(a,b;x)}\)
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 \(x\ge 0\), the Catalan–Qi function C(a, b; x) has integral representations
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. \(\square\)
Remark 5
Letting \(a=\frac{1}{2}\), \(b=2\), and \(x=n\) in (22) and (21) respectively reduce to the first integral representation in (9) and its equivalent form (8).
Remark 6
In https://en.wikipedia.org/wiki/Catalan_number, it was said that the integral representation (8) means that the Catalan numbers \(C_n\) are a solution of the Hausdorff moment problem on the interval [0, 4] instead of [0, 1]. Analogously, we guess that the integral representation (21) probably means that the Catalan–Qi numbers C(a, b; n) are a solution of the Hausdorff moment problem on the interval \(\bigl [0,\frac{b}{a}\bigr ]\) instead of [0, 1] and [0, 4].
Recall from Mitrinović et al. (1993, Chapter XIII), Schilling et al. (2012, Chapter 1), and Widder (1941, Chapter IV) that an infinitely differentiable function f is said to be completely monotonic on an interval I if it satisfies \(0\le (-1)^kf^{(k)}(x)<\infty\) on I for all \(k\ge 0\). It is known (Widder 1941, p. 161, Theorem 12b) that a function f is completely monotonic on \((0,\infty )\) if and only if it is a Laplace transform \(f(t)=\int _0^\infty e^{-ts}{{\text{d}}}\mu (s)\) of a positive measure \(\mu\) defined on \([0,\infty )\) such that the above integral converges on \((0,\infty )\).
Theorem 5
For \(b>a>0\), we have
where
is the falling factorial. Consequently, the function
for \(N\in \{0\}\cup {\mathbb {N}}\) and \(b>a>0\) is completely monotonic in \(x\in [0,\infty )\), where \(\lfloor x\rfloor\) denotes the floor function whose value is the largest integer less than or equal to x.
Proof
The integral representation (21) can be rearranged as
Further utilizing the well-known power series expansion
arrives at
which can be reformulated as (23).
Rewriting (23) as
considering the non-negativity of \((-1)^{k-\lfloor b-a\rfloor }\langle b-a-1\rangle _k\), and employing the complete monotonicity of \(\frac{1}{x+a+k}\) in \(x\in [0,\infty )\) reveal the complete monotonicity of the function (24). The proof of Theorem 5 is complete. \(\square\)
Remark 7
When taking \(a=\frac{1}{2}\) and \(b=2\), Theorem 5 becomes a part of conclusions in Qi (2015a, Theorem 1.1).
Logarithmically complete monotonicity of the Catalan–Qi function \(\varvec{C(a,b;x)}\)
An infinitely differentiable and positive function f is said to be logarithmically completely monotonic on an interval I if \(0\le (-1)^k[\ln f(x)]^{(k)}<\infty\) hold on I for all \(k\in {\mathbb {N}}\). The inclusions
were discovered in Berg (2004), Guo and Qi (2010), Qi and Chen (2004), Qi and Guo (2004), where \({\mathcal {L}[I]}\), \({\mathcal {C}[I]}\), and \({\mathcal {S}}\) denote respectively the set of all logarithmically completely monotonic functions on an interval I, the set of all completely monotonic functions on I, and the set of all Stieltjes transforms. See also the monograph Schilling et al. (2012) and plenty of references therein.
Recall from monographs Mitrinović et al. (1993, pp. 372–373) and Widder (1941, p. 108, Definition 4) that a sequence \(\{\mu _n\}_{0\le n\le \infty }\) is said to be completely monotonic if its elements are non-negative and its successive differences are alternatively non-negative, that is,
for \(n,k\ge 0\), where
Recall from Widder (1941, p. 163, Definition 14a) that a completely monotonic sequence \(\{a_n\}_{n\ge 0}\) is minimal if it ceases to be completely monotonic when \(a_0\) is decreased.
Theorem 6
The function
is logarithmically completely monotonic on \((0,\infty )\) if and only if \(a\gtrless b\). Consequently, the sequence
is completely monotonic, minimal, and logarithmically convex.
Proof
In Qi and Li (2015, Theorem 1.1), it was proved that, when \(a\gtrless b\), the function
for \(c>0\) is logarithmically completely monotonic on \([0,\infty )\) if and only if \(c\gtreqless \frac{\Gamma (b)}{\Gamma (a)}\). It is easy to see that
Therefore, the function \({\mathcal {C}}^{\pm 1}(a,b;x)\) is logarithmically completely monotonic on \([0,\infty )\) if and only if \(a\gtrless b\). Consequently, the function \({\mathcal {C}}^{-1}\bigl (\frac{1}{2},2;x\bigr )\) is logarithmically completely monotonic, and then completely monotonic and logarithmically convex, on \([0,\infty )\). 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 \(0\le x<\infty\) such that \(f(n)=a_n\) for \(n\ge 0\) is that \(\{a_n\}_0^\infty\) should be a minimal completely monotonic sequence. The proof of Theorem 6 is complete. \(\square\)
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 \(x\ge 0\). Then
-
1.
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 \(x\in [0,\infty )\);
-
2.
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 \(x\in [0,\infty )\);
where \(x_0\) is the unique zero of the equation
and satisfies \(x_0\in \bigl (0,\frac{1}{2}\bigr )\). Consequently, the Catalan numbers \(C_n\) for \(n\in {\mathbb {N}}\) 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 \((0,\infty )\) if and only if \(\alpha \le 1\). This means that
that is,
This can also be verified by virtue of the inequality
which is a special case of Guo and Qi (2010, Lemma 3), and by virtue of the equality
Since the function \(\psi (x+b)-\psi (x+a)\) is increasing (or decreasing, respectively) if and only if \(b<a\) (or \(b>a\), respectively) and
for all \(a,b>0\), we obtain that for all \(a,b>0\) with \(a\ne b\) the function \(\frac{\psi (x+b)-\psi (x+a)}{\ln b-\ln a}\) is strictly decreasing on \([0,\infty )\) and
It is clear that the first derivative
if and only if
which can be rewritten as
and
As a result, considering (30) and (31), we see that the Catalan–Qi function C(a, b; x) for all \(a,b>0\) with \(a\ne b\) is not monotonic on \([0,\infty )\) and that
-
1.
when \(b>a\), the function C(a, b; x) is decreasing in \(x\in (0,x_0)\) and increasing in \(x\in (x_0,\infty )\);
-
2.
when \(b<a\), the function C(a, b; x) is increasing in \(x\in (0,x_0)\) and decreasing in \(x\in (x_0,\infty )\);
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
where \(\lim _{u\rightarrow 0^+}\bigl [u-\frac{1}{\psi '(u)}\bigr ]=0\) and
Employing the asymptotic expansion
in Abramowitz and Stegun (1972, p. 260, 6.4.11) yields
Due to \([\psi '(x)]^2+\psi ''(x)>0\) on \((0,\infty )\), see Alzer (2004), Qi (2015b), Qi and Li (2015), Qi et al. (2013) and plenty of closely-related references therein, the function \(u-\frac{1}{\psi '(u)}\) is strictly increasing, and so
on \((0,\infty )\). Accordingly, the unique zero \(x_0\) of the Eq. (29) belongs to \(\bigl (0,\frac{1}{2}\bigr )\).
It is immediate that
Since the tri-gamma function \(\psi '(x)\) is completely monotonic on \((0,\infty )\), inequalities
for \(k\in {\mathbb {N}}\) hold if and only if \(b\lessgtr a\). The proof of Theorem 7 is complete. \(\square\)
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 \([0,\infty )\).
Proof
By (6), it follows that
which can be straightforwardly verified to be a logarithmically completely monotonic function on \([0,\infty )\). By the first inclusion in (26), we obtain the required complete monotonicity of the function (32). \(\square\)
Remark 10
The integral representation (22) can be rewritten as
for \(b>a>0\) and \(x\ge 0\). 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 \([0,\infty )\). This conclusion is weaker than Theorem 8.
Theorem 9
For \(b>a>0\), the function
is logarithmically completely monotonic on \([0,\infty )\).
Proof
This follows from the integral representation (7). \(\square\)
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).
A generating function of the Catalan–Qi sequence \(\varvec{C(a,b;n)}\)
In this section, we discover that \({}_2F_1\bigl (a,1;b;\frac{bt}{a}\bigr )\) is a generating function of the Catalan–Qi numbers C(a, b; n).
Theorem 10
For \(a,b>0\) and \(n\ge 0\), the Catalan–Qi numbers C(a, b; n) can be generated by
and, conversely, satisfy
Proof
Using the relation \((z)_n \Gamma (z)=\Gamma (z+n)\) for \(n \ge 0\), we have
As a result, we obtain
Using the relation \((-n)_{n+i}=0\) for \(i\in {\mathbb {N}}\), 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. \(\square\)
Remark 12
(An alternative proof of (33) for \(b>1\)) In Abramowitz and Stegun (1972, p. 558, 15.3.1), 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 \(b>1\).
Remark 13
Combining (2) and (34) brings out
A double inequality of the Catalan–Qi function \(\varvec{C(a,b;x)}\)
Finally we present a double inequality of the Catalan–Qi function C(a, b; x).
Theorem 11
Let \(B_i\) for \(i\in {\mathbb {N}}\) be the Bernoulli numbers defined by (17) and let I be the exponential mean defined by (19). Then the Catalan–Qi function C(a, b; x) satisfies the double inequality
Consequently, we have
Proof
In Koumandos (2006, Theorem 3), it was obtained that
for \(m\in {\mathbb {N}}\) and \(x>0\). Substituting this double inequality into the integral representation (7) and straightforward computing lead to the double inequality (36).
The double inequality (37) follows from letting \(x\rightarrow 0\) in (36) and simplifying. The proof of Theorem 11 is complete. \(\square\)
Remark 14
The double inequality (36) generalizes a double inequality in Qi (2015a, Theorem 1.2).
Conclusions
The main conclusions of this paper are stated in Theorems 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. Concretely speaking, a new expression, several asymptotic expansions, several integral representations, logarithmic convexity, complete monotonicity, minimality, logarithmically complete monotonicity, a generating function, and several inequalities of the Catalan numbers, the Catalan function, and the Catalan–Qi function are presented and an exponential expansion and a double inequality for the ratio of two gamma functions are derived. These conclusions generalize and extend some known results. More importantly, these conclusions provide new viewpoints of understanding and supply new methods of investigating the Catalan numbers in combinatorics and number theory. Moreover, these conclusions connect the Catalan numbers with the ratios of two gamma functions in the theory of special functions. In other words, the main conclusions in this paper will deepen and promote the study of the Catalan numbers and related concepts in combinatorics and number theory.
Remark 15
This paper is a companion of the articles Liu et al. (2015), Mahmoud and Qi (2016), Qi (2015a, d, e), Qi and Guo (2016a, b), Qi et al. (2015b, c, d, e), Shi et al. (2015) and a revised version of the preprint Qi et al. (2015a).
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The authors appreciate anonymous referees for their careful corrections to and valuable comments on the original version of this paper.
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Qi, F., Mahmoud, M., Shi, XT. et al. Some properties of the Catalan–Qi function related to the Catalan numbers. SpringerPlus 5, 1126 (2016). https://doi.org/10.1186/s40064-016-2793-1
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DOI: https://doi.org/10.1186/s40064-016-2793-1
Keywords
- Property
- Catalan number
- Catalan function
- Catalan–Qi function
- Asymptotic expansion
- Integral representation
- Logarithmic convexity
- Complete monotonicity
- Logarithmically complete monotonicity
- Minimality
- Inequality
- Ratio of gamma functions