An integral representation, complete monotonicity ...

AN INTEGRAL REPRESENTATION, COMPLETE MONOTONICITY, AND INEQUALITIES OF THE CATALAN NUMBERS FENG QI, XIAO-TING SHI, AND FANG-FANG LIU

Abstract. In the paper, by virtue of the Cauchy integral formula in the theory of complex functions, the authors establish an integral representation for the generating function of the Catalan numbers in combinatorics. From this, the authors derive an alternative integral representation, complete monotonicity, determinantal and product inequalities for the Catalan numbers.

1. Introduction It is known [40, 34] in combinatorial analysis that the Catalan numbers Cn 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 Cn−2 . The Catalan numbers Cn can be generated by √ ∞ ∑ 1 1 − 1 − 4x √ = = Cn xn 2x 1 + 1 − 4x n=0 = 1 + x + 2x2 + 5x3 + 14x4 + 42x5 + 132x6 + 429x7 + 1430x8 + · · ·

(1.1)

Explicit formulas of Cn for n ≥ 0 include ( ) 1 2n (2n)! 2n (2n − 1)!! Cn = = = n+1 n n!(n + 1)! (n + 1)! ( 1 ) ( ) 1 2n 2 = (−1)n 22n+1 = = 2 F1 (1 − n, −n; 2; 1) n+1 n n−1 and 4n Γ(n + 1/2) Cn = √ , (1.2) π Γ(n + 2) where ∫ ∞ Γ(z) = tz−1 e−t d t, ℜ(z) > 0 0

is the classical Euler gamma function and p Fq (a1 , . . . , ap ; b1 , . . . , bq ; z)

=

∞ ∑ (a1 )n · · · (ap )n z n (b1 )n · · · (bq )n n! n=0

2010 Mathematics Subject Classification. Primary 11R33; Secondary 05A20, 11B75, 11B83, 11R33, 11S23, 26A48, 30E20, 33B15, 44A10, 44A20. Key words and phrases. Catalan number; generating function; integral representation; complete monotonicity; determinantal inequality; product inequality; Cauchy integral formula. This paper was typeset using AMS-LATEX. 1

2

F. QI, X.-T. SHI, AND F.-F. LIU

is the generalized hypergeometric series defined for complex numbers ai ∈ C and bi ∈ C \ {0, −1, −2, . . . }, for positive integers p, q ∈ N, and in terms of the rising factorials { x(x + 1)(x + 2) · · · (x + n − 1), n ≥ 1, (x)n = 1, n = 0. An integral representation 1 Cn = 2π

∫ 0

4

√

4−x n x dx x

for the Catalan numbers was listed in [40]. In [3, 37], the asymptotic expansion ( ) 4x 1 9 1 145 1 Cx ∼ √ − + + · · · 8 x5/2 128 x7/2 π x3/2 for the Catalan numbers Cn was given. Recently, among other things, the formula ( ) n−1 n k ∏ 2n ∑ 1 ∑ ℓ k Cn = (−1)n (−1) (ℓ − 2m) k n! 2 ℓ m=0 k=0 ℓ=0 ( ) n 2n ∑ k! 2n − k − 1 = [2(n − k) − 1]!!. n! 2k 2(n − k) k=0

was found in [29, Theorem 3]. For more information on the Catalan numbers Cn , please refer to the monographs [1, 3] and references cited therein. In the paper [33], motivated by the explicit expression (1.2) and by virtue of an integral representation of the gamma function Γ(x), the authors established an integral representation of the Catalan numbers Cx for x ≥ 0. Theorem 1.1 ([33, Theorem 1]). For x ≥ 0, we have ) ] [∫ ∞ ( 1 1 e−t/2 − e−2t −xt e3/2 4x (x + 1/2)x 1 √ − + e d t . (1.3) Cx = exp et − 1 t 2 t π (x + 2)x+3/2 0 Recall from [10, Chapter XIII], [32, Chapter 1], and [39, 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 [20, 21] 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) < ∞ hold on I for all k ∈ N. For more information on logarithmically completely monotonic functions, please refer to [22, 23, 26, 30]. The formula (1.3) can be rearranged as [√ ] ∫ ∞ ( ) ) 1 1 1 1 ( −t/2 π (x + 2)x+3/2 ln 3/2 x C = − + e − e−2t e−xt d t. (1.4) x t x t e −1 t 2 e 4 (x + 1/2) 0 ( 1 ) 1 1 1 Since the function t et −1 − t + 2 is positive on (0, ∞), see [6, 27, 41] and references therein, the right hand side of (1.4) is a completely monotonic function on (0, ∞). This means that the function (x + 2)x+3/2 Cx (1.5) 4x (x + 1/2)x

AN INTEGRAL REPRESENTATION OF THE CATALAN NUMBERS

3

is logarithmically completely monotonic on (0, ∞). Because any logarithmically completely monotonic function must be completely monotonic, see [23, Eq. (1.4)] and references therein, the function (1.5) is also completely monotonic on (0, ∞). By virtue of (1.2), the function (1.5) can be rewritten as (x + 2)x+3/2 Γ(x + 1/2) , (x + 1/2)x Γ(x + 2)

x > 0.

(1.6)

Hence, the logarithmically complete monotonicity of (1.5) implies the logarithmically complete monotonicity of (1.6). The function (1.6) is a special case F1/2,2 (x) of the general function Fa,b (x) =

Γ(x + a) (x + b)x+b−a , (x + a)x Γ(x + b)

a, b ∈ R,

a ̸= b x > − min{a, b}.

(1.7)

We note that the function Fa,b (x) does not appear in the expository and survey articles [14, 15, 23, 24, 25] and plenty of references therein. Therefore, it is significant to naturally pose an open problem below. Open Problem 1.1 ([33, Open Problem 1]). What are the necessary and sufficient conditions on a, b ∈ R such that the function Fa,b (x) defined by (1.7) is (logarithmically) completely monotonic in x ∈ (− min{a, b}, ∞)? This problem was answered respectively in [12, 28] as follows. Theorem 1.2 ([12, Theorem 2]). The sufficient conditions on a, b such that the function [Fa,b (x)]±1 defined by (1.7) is logarithmically completely monotonic in x ∈ (− min{a, b}, ∞) are (a, b) ∈ D± (a, b), where { } 1 D± (a, b) = {a ≥ 1, a ≷ b} ∪ a ≤ , a ≶ b . 2 The necessary conditions on a, b for the function [Fa,b (x)]±1 to be logarithmically completely monotonic in x ∈ (− min{a, b}, ∞) are a−b a(a − b) R . 2 Theorem 1.3 ([28, Theorem 3]). For a, b > 0, the function Fa,b (x) defined by (1.7) has the exponential representation [ ) ] ∫ ∞ ( ( −at ) 1 1 1 −bt −xt Fa,b (x) = exp b − a + − −a e −e e dt (1.8) t 1 − e−t t 0 on [0, ∞) and the function [Fa,b (x)]±1 is logarithmically completely monotonic on [0, ∞) if and only if (a, b) ∈ D± (a, b). Recall from monographs [10, pp. 372–373] and [39, p. 108, Definition 4] that a sequence {µn }0≤n≤∞ is said to be completely monotonic if its elements are nonnegative and its successive differences are alternatively non-negative, that is (−1)k ∆k µn ≥ 0 for n, k ≥ 0, where ∆k µn =

k ∑ m=0

(−1)m

( ) k µn+k−m . m

Recall from [39, p. 163, Definition 14a] that a completely monotonic sequence {an }n≥0 is minimal if it ceases to be completely monotonic when a0 is decreased.

4

F. QI, X.-T. SHI, AND F.-F. LIU

Let λ = (λ1 , λ2 , . . . , λn ) ∈ Rn and µ = (µ1 , µ2 , . . . , µn ) ∈ Rn . A sequence λ is said to be majorized by µ (in symbols λ ≼ µ) if k ∑ ℓ=1

λ[ℓ] ≤

k ∑

µ[ℓ] ,

k = 1, 2, . . . , n − 1 and

ℓ=1

n ∑ ℓ=1

λℓ =

n ∑

µℓ ,

ℓ=1

where λ[1] ≥ λ[2] ≥ · · · ≥ λ[n] and µ[1] ≥ µ[2] ≥ · · · ≥ µ[n] are respectively the components of λ and µ in decreasing order. A sequence λ is said to be strictly majorized by µ (in symbols λ ≺ µ) if λ is not a permutation of µ. For example, ( ) ( ) ( ) 1 1 1 1 1 1 ≺ ,..., ,..., ,0 ≺ ,..., , 0, 0 ≺ · · · n−1 n−1 n−2 n−2 |n {z n} | {z } | {z } n

n−1

n−2

(

) ( ) 1 1 1 1 1 ≺ , , , 0, . . . , 0 ≺ , , 0, . . . , 0 ≺ (1, 0, . . . , 0). 3 3 3 2 2 For more information on the theory of majorization and its applications, please refer to monographs [5, 7] and closely related references therein. In this paper, by virtue of the Cauchy integral formula in the theory of complex functions, we establish an integral representation for the generating function √1 of the Catalan numbers Cn . From this, we derive an alternative integral 1+ 1−4x representation, complete monotonicity, determinantal {and }product inequalities for the Catalan numbers Cn for n ≥ 0 and the sequences C4nn n≥0 and {n!Cn }n≥0 . Our main results in this paper can be stated as the following theorems. ( ) Theorem 1.4. For x ∈ −∞, 14 , we have √ ∫ 1 1 ∞ t √ = d t. (1.9) π 0 (t + 1/4)(t − x + 1/4) 1 + 1 − 4x Consequently, the Catalan numbers Cn for n ≥ 0 can be represented by √ ∫ ∫ 1 ∞ t 2 ∞ t2 Cn = d t = d t. (1.10) n+2 2 π 0 (t + 1/4) π 0 (t + 1/4)n+2 { } and the sequence C4nn n≥0 is completely monotonic and minimal. If m ≥ 1 and a0 , a1 , . . . , am be nonnegative integers, then ( )m−1 C∑ m ∏ m Ca0 Ca0 +ak ak ∑k=0 ≥ (1.11) m 4a0 4a0 +ak 4 k=0 ak k=1

and where |ekj |m

Cai +aj 4ai +aj ≥ 0, m denotes a determinant of order m with elements ekj .

(1.12)

Theorem 1.5. Let m ∈ N and let n and ak for 1 ≤ k ≤ m be nonnegative integers. Then the Catalan numbers Cn satisfy (−1)ai +aj Cn+ai +aj ≥ 0 (1.13) m and

Cn+ai +aj ≥ 0, m

(1.14)

where Cℓ = ℓ!Cℓ ,

ℓ ≥ 0.

(1.15)

AN INTEGRAL REPRESENTATION OF THE CATALAN NUMBERS

5

Theorem 1.6. Let m ∈ N and let λ and µ be two m-tuples of nonnegative integers such that λ ≼ µ. Then m m ∏ ∏ (1.16) Cn+λi ≤ Cn+µi , i=1

i=1

where Cℓ is defined by (1.15). Consequently, (1) the infinite sequence {Cn }n≥0 is logarithmically convex, (2) the inequality n k Cℓ+k ≤ Cℓ+n Cℓn−k is valid for ℓ ≥ 0 and n > k > 0.

(1.17)

Theorem 1.7. If ℓ ≥ 0, n ≥ k ≥ m, k ≥ n − k, and m ≥ n − m, then (ℓ + m)!(ℓ + n − m)! Cℓ+k Cℓ+n−k ≥ . Cℓ+m Cℓ+n−m (ℓ + k)!(ℓ + n − k)!

(1.18)

For n, m ∈ N and ℓ ≥ 0, let Gn,m,ℓ = Cℓ+n+2m (Cℓ )2 − Cℓ+n+m Cℓ+m Cℓ − Cℓ+n Cℓ+2m Cℓ + Cℓ+n (Cℓ+m )2 , Hn,m,ℓ = Cℓ+n+2m (Cℓ )2 − 2Cℓ+n+m Cℓ+m Cℓ + Cℓ+n (Cℓ+m )2 , In,m,ℓ = Cℓ+n+2m (Cℓ )2 − 2Cℓ+n Cℓ+2m Cℓ + Cℓ+n (Cℓ+m )2 , where Cℓ is defined by (1.15). Then Gn,m,ℓ ≥ 0,

Hn,m,ℓ ≥ 0,

Hn,m,ℓ Q Gn,m,ℓ

when m ≶ n,

(1.19) (1.20)

and In,m,ℓ ≥ Gn,m,ℓ ≥ 0

when n ≥ m.

(1.21)

2. Proofs of Theorems 1.4 to 1.7 We are now in a position to prove Theorems 1.4 to 1.7. Proof of Theorem 1.4. As usual, we use ln x for the logarithmic function having base e and applied to the positive argument x > 0. Further, the principal branch of the holomorphic extension of ln x from the open half-line (0, √ ∞) to the cut plane C \ (−∞, 0] is denoted by ln z = ln |z| + i arg z, where i = −1 is the imaginary unit and the principal value arg z of the argument of z satisfies | arg z| < π. Let [ ) ( ) 2 1 1 f (z) = , z ∈ C \ , ∞ , arg z − ∈ (0, 2π) 4 4 1 + exp ln(1−4z) 2

and

2

, z ∈ C \ [0, ∞), arg z ∈ (0, 2π). 1 + exp ln(−4z) 2 By virtue of the Cauchy integral formula in the theory of complex functions, for any fixed point z0 = x0 + iy0 ∈ C \ [0, ∞), we have ∫ 1 F (ξ) F (z0 ) = d ξ, 2πi L ξ − z0 F (z) =

where L is a positively oriented contour L(r, R) in C \ [0, ∞), as showed in Figure 1, such that

6

F. QI, X.-T. SHI, AND F.-F. LIU

{ (1) (2) (3) (4) (5)

0 < r < |y0 | ≤ |z0 | < R, y0 ̸= 0, 0 < r < −x0 = |z0 | < r, y0 = 0, ] [ L(r, R) consists of the half circle z = reiθ for θ ∈ π2 , 3π 2 , L(r, R) consists of the half lines z = x ± ir for x > 0, L(r, R) consists of the circle |z| = R, the half lines z = x ± ir √ for x ≥ 0 cut the circle |z| = R at the points R(r) ± ir and 0 < R(r) = R2 − r2 → R as r → 0.

y

r R

r

R(r) R x

Figure 1. The positively oriented contour L(r, R) in C \ [0, ∞) Then the integral on the circle with radius R equals ∫ 2π−arcsin(r/R) 1 2Rieiθ ( )[ iθ ) ] d θ 2πi arcsin(r/R) Reiθ − z0 1 + exp ln(−4Re 2 ∫ 1 2π−arcsin(r/R) 1 = ( )[ iθ ) ] d θ z π arcsin(r/R) 1 − Re0iθ 1 + exp ln(−4Re 2 ∫ 1 2π−arcsin(r/R) 1 = ( )[ iθ ) ] d θ z0 π arcsin(r/R) 1 − Reiθ 1 + exp ln(4R)+i arg(−4Re 2 which tends uniformly to 0 as R → ∞. [ ] Further, the integral on the half circle z = reiθ for θ ∈ π2 , 3π is 2 ∫ 3π/2 iθ 2rie 1 − ( )[ iθ ) ] d θ iθ 2πi π/2 re − z0 1 + exp ln(−4re 2

AN INTEGRAL REPRESENTATION OF THE CATALAN NUMBERS

1 =− π =− =−

1 π 1 π

∫

3π/2

π/2

∫

z0 reiθ

1 )[ iθ ) ] d θ 1 + exp ln(−4re 2

( 1−

z0 reiθ

1 )[ iθ ) ] d θ 1 + exp ln(4r)+i arg(−4re 2

( 1−

z0 reiθ

1 )[ √ iθ ) ] d θ 1 + 2 r exp i arg(−4re 2

3π/2

π/2

∫

( 1−

3π/2

π/2

which tends uniformly to 0 as r → 0+ . Continuously, because 2

F (x + ir) = 1 + exp

ln(−4x−4ri) 2

2

= 1 + exp 2

√ ln(4 x2 +r 2 )+i[arctan(r/x)−π] 2

√ [ ] 1 + 2 4 x2 + r2 cos arctan(r/x)−π + i sin arctan(r/x)−π 2 2 2 = √ [ ] arctan(r/x) 4 2 2 1 + 2 x + r sin − i cos arctan(r/x) 2 2 2 =[ √ √ ] 1 + 2 4 x2 + r2 sin arctan(r/x) − i2 4 x2 + r2 cos arctan(r/x) 2 2 √ √ ( arctan(r/x) ) arctan(r/x) 4 4 2 2 2 2 2 1 + 2 x + r sin + i4 x + r cos 2 2 =[ √ [ √ ] arctan(r/x) ]2 4 4 2 2 2 + r 2 cos arctan(r/x) 2 1 + 2 x + r sin + 2 x 2 2 √ 2 + i4 x → , r → 0+ , 1 + 4x =

the integral on the half lines z = x ± ir for x > 0 is equal to [∫ ] ∫ 0 R(r) 1 F (x + ir) F (x − ir) dx + dx 2πi 0 x + ir − z0 R(r) x − ir − z0 ∫ R(r) 1 (x − ir − z0 )F (x + ir) − (x + ir − z0 )F (x − ir) = dx 2πi 0 (x + ir − z0 )(x − ir − z0 ) ∫ R(r) 1 (x − z0 )[F (x + ir) − F (x − ir)] − ir[F (x + ir) + F (x − ir)] = dx 2πi 0 (x + ir − z0 )(x − ir − z0 ) )] [ ( )] [ ( ∫ R(r) (x − z0 ) F (x + ir) − F x + ir − ir F (x + ir) + F x + ir 1 = dx 2πi 0 (x + ir − z0 )(x − ir − z0 ) [ ] [ ] ∫ R(r) (x − z0 ) F (x + ir) − F (x + ir) − ir F (x + ir) + F (x + ir) 1 dx = 2πi 0 (x + ir − z0 )(x − ir − z0 ) ∫ R(r) 1 (x − z0 )[2iℑ(F (x + ir))] − ir[2ℜ(F (x + ir))] = dx 2πi 0 (x + ir − z0 )(x − ir − z0 ) √ ∫ ∞ 1 2i 4 x → d x, r → 0+ , R → ∞ 2πi 0 x − z0 1 + 4x √ ∫ 4 ∞ x = d x. π 0 (1 + 4x)(x − z0 )

7

8

F. QI, X.-T. SHI, AND F.-F. LIU

Consequently, it follows that 2 0) 1 + exp ln(−4z 2

4 = π

∫ 0

∞

√ x dx (1 + 4x)(x − z0 )

(2.1)

for z0 ∈ C \ [0, ∞) and arg z0 ∈ (0, 2π). Furthermore, since F (z) = f (z + 14 ) and the point z0 in (2.1) is arbitrary, making use of (2.1) arrives at √ ∫ 4 ∞ 1 x = dx (2.2) ln(1−4z) π 0 (1 + 4x)(x − z + 1/4) 1 + exp 2 [1 ) ( ) 1 for ( z ∈1 )C \ 4 , ∞ and arg z − 4 ∈ (0, 2π). In particular, when taking z = x ∈ −∞, 4 , the formula (2.2) becomes the integral representation (1.9). Differentiating n ≥ 0 times with respect to x on both sides of (1.9) and taking the limit x → 0 yield √ ) ( ∫ ∞ dn 1 4 n! t √ lim = lim dt x→0 d xn 1 + π x→0 0 (1 + 4t)(t − x + 1/4)n+1 1 − 4x √ ∫ ∞ t 4 = n! dt π (1 + 4t)(t + 1/4)n+1 0 (2.3) √ ∫ ∞ t 1 = n! dt π (t + 1/4)n+2 0 ∫ ∞ 2 s2 = n! d s. π (s2 + 1/4)n+2 0 As a result, by virtue of (1.1), the integral representations in (1.10) follow. We can rewrite the first integral representation in (1.10) as √ ∫ t Cn 1 ∞ = d t, n ≥ 0. n+2 4 π 0 (1 + 4t)n+2 It is easy to see that the function ax for a < 1 is completely monotonic in x ∈ 1 [0, ∞). Hence, the function (1+4t) x+2 for any fixed t > 0 is completely monotonic in x ∈ [0, ∞). Equivalently, the function C4xx is completely monotonic in x ∈ [0, ∞). Theorem 14b in [39, p. 164] states that “a necessary and sufficient condition that there should exist a function f (x) completely monotonic in 0 ≤ x < ∞ such that f (n) = an for n ≥ 0 is that {an }∞ 0 should be a minimal completely monotonic sequence”. As a result, the sequence C4nn for n ≥ 0 is minimal completely monotonic. In [8] and [10, pp. 369 and 374], it was obtained that if f is completely monotonic on [0, ∞) and m ≥ 1, then (m ) m ∑ ∏ m−1 [f (x0 )] f xk ≥ f (x0 + xk ) (2.4) k=0

k=1

and |f (xi + xj )|m ≥ 0. Now we consider the function √ ∫ 16 ∞ t d t, h(x, s) = π 0 (1 + 4t)(4x + 4t + 1)s+1

(2.5)

x, s ≥ 0.

AN INTEGRAL REPRESENTATION OF THE CATALAN NUMBERS

9

It is easy to see that the function h(x, s) for any fixed x > 0 is completely monotonic in s ∈ [0, ∞). By virtue of (2.3), we obtain Cai , 4ai where ai for i ≥ 0 are nonnegative integers. Replacing the function f and nonnegative numbers x0 , x1 , . . . , xm in (2.4) and (2.5) by the function h(x, s) and nonnegative integers a0 , a1 , . . . , am respectively yields ( m ) m ∑ ∏ m−1 [h(x, a0 )] h x, ak ≥ h(x, a0 + ak ) (2.6) lim h(x, ai ) =

x→0+

k=0

k=1

and |h(x, ai + aj )|m ≥ 0.

(2.7)

Therefore, taking x → 0 in (2.6) and (2.7) leads to (1.11) and (1.12). The proof of Theorem 1.4 is complete.  +

Proof of Theorem 1.5. From (2.3), we observe that Cn = lim hn (x), x→0

where 4 hn (x) = π

∫

√

∞

t dt (1 + 4t)(t − x + 1/4)n+1 0 ) ( and hn (x) is absolutely monotonic on −∞, 41 . This means that the function √ ∫ 4 ∞ t Hn (x) = hn (−x) = dt (2.8) π 0 (1 + 4t)(x + t + 1/4)n+1 ) ( is completely monotonic on − 14 , ∞ ⊃ [0, ∞) and that Cn = lim Hn (x). x→0

A direct computation gives (n + k)! (n + k)! Cn+k Hn+k (x) → (−1)k Cn+k = (−1)k (2.9) n! n! n! for k ≥ 0 as x → 0. In the paper [9], or see [10, p. 367], it was obtained that if f is a completely monotonic function on [0, ∞), then (a +a ) f i j (x) ≥ 0 (2.10) m Hn(k) (x) = (−1)k

and

(−1)ai +aj f (ai +aj ) (x) ≥ 0. m

Applying f in (2.10) and (2.11) to the function Hn (x) yields (a +a ) Hn i j (x) ≥ 0 m and

(−1)ai +aj Hn(ai +aj ) (x)

m

≥ 0.

Letting x → 0 in (2.12) and (2.13) and making use of (2.9) arrive at (−1)ai +aj Cn+ai +aj ≥ 0 n! m

(2.11)

(2.12) (2.13)

(2.14)

10

F. QI, X.-T. SHI, AND F.-F. LIU

Cn+ai +aj ≥ 0. (2.15) n! m Further simplifying (2.14) and (2.15) leads to (1.13) and (1.14). The proof of Theorem 1.5 is complete.  and

Proof of Theorem 1.6. In [10, p. 367, Theorem 2] and [36, p. 106, Theorem A], which are a minor correction of [2, Theorem 1], it was obtained that if f is a completely monotonic function on (0, ∞) and λ ≼ µ, then n n ∏ ∏ (2.16) f (λi ) (x) ≤ f (µi ) (x) . i=1

i=1

The equality in (2.16) is valid only when λ and µ are identical or when f (x) = e−cx for c ≥ 0. Applying the inequality (2.16) to Hn (x) defined in (2.8) creates m m ∏ ∏ (λi ) (µi ) Hn (x) ≤ Hn (x) . i=1

i=1

Taking the limit x → 0 on both sides of the above inequality and making use of (2.9) reveal m m ∏ ∏ λi Cn+λi µi Cn+µi (−1) ≤ (−1) n! n! i=1

i=1

which is equivalent to (1.16). From the majorization relation (i + 2, i) ≽ (i + 1, i + 1) for i ≥ 0 and the inequality (1.16), the logarithmic convexity of the sequence {Cn }n≥0 follows immediately. In [10, p. 369] and [11, p. 429, Remark], it was formulated that if f (t) is a completely monotonic function such that f (k) (t) ̸= 0 for k ≥ 0, then the sequence [ ] si (t) = ln (−1)i−1 f (i−1) (t) , i ≥ 1 is convex. Applying this result to the function Hn (x) and making use of (2.9) figures out that the sequence ] [ Cn+i−1 ln (−1)i−1 Hn(i−1) (x) → ln , x→0 n! for i ≥ 1 is convex. Hence, the sequence {Cn }n≥0 is logarithmically convex. As did in [2], considering the majorization relation (k, k, . . . , k ) ≺ (n, . . . , n, 0, . . . , 0) | {z } | {z } | {z } n

k

n−k

for n > k, the inequality (2.16) becomes [ ]n [ ]k (−1)nk f (k) (x) ≤ (−1)nk f (n) (x) [f (x)]n−k ,

n > k > 0.

Substituting Hℓ (x) for f (x) in the above inequality, letting t → 0, and utilizing (2.9) procure [ (k) ]n [ (n) ]k (−1)nk Hℓ (x) ≤ (−1)nk Hℓ (x) [Hℓ (x)]n−k which is equivalent to [ ]n ]k [ nk k Cℓ+k nk n Cℓ+n (−1) (−1) ≤ (−1) (−1) (Cℓ )n−k ℓ! ℓ!

AN INTEGRAL REPRESENTATION OF THE CATALAN NUMBERS

11

for n > k > 0 and ℓ ≥ 0. This may be simplified as (1.17). The proof of Theorem 1.6 is complete.  Proof of Theorem 1.7. In [35, p. 397, Theorem D], it was recovered that if f (x) is completely monotonic on (0, ∞) and if n ≥ k ≥ m, k ≥ n − k, and m ≥ n − m, then (−1)n f (k) (x)f (n−k) (x) ≥ (−1)n f (m) (x)f (n−m) (x). Replacing f (x) by the function Hℓ (x) defined by (2.8) in the above inequality leads to (k) (n−k) (m) (n−m) (−1)n Hℓ (x)Hℓ (x) ≥ (−1)n Hℓ (x)Hℓ (x). Further taking x → 0 and employing (2.9) find Cℓ+k Cℓ+n−k Cℓ+m Cℓ+n−m (−1)n−k ≥ (−1)n (−1)m (−1)n−m . ℓ! ℓ! ℓ! ℓ! Simplifying this inequality leads to (1.18). In [36, Theorem 1 and Remark 2], it was obtained that if f is completely monotonic on (0, ∞) and { [ ]2 } Gn,m = (−1)n f (n+2m) f 2 − f (n+m) f (m) f − f (n) f (2m) f + f (n) f (m) , { [ ]2 } Hn,m = (−1)n f (n+2m) f 2 − 2f (n+m) f (m) f + f (n) f (m) , { [ ]2 } In,m = (−1)n f (n+2m) f 2 − 2f (n) f (2m) f + f (n) f (m) (−1)n (−1)k

for n, m ∈ N, then Gn,m ≥ 0, Hn,m ≥ 0, and Hn,m Q Gn,m In,m ≥ Gn,m ≥ 0

when m ≶ n, when n ≥ m.

Replacing f (x) by Hℓ (x) in Gn,m , Hn,m , and In,m , taking x → 0, and employing (2.9) produce { (n+2m) 2 (n+m) (m) (n) (2m) (n) [ (m) ]2 } Hℓ Hℓ − Hℓ Hℓ Hℓ + Hℓ Hℓ Gn,m = (−1)n Hℓ Hℓ − Hℓ ] 1 [ = Cℓ+n+2m (Cℓ )2 − Cℓ+n+m Cℓ+m Cℓ − Cℓ+n Cℓ+2m Cℓ + Cℓ+n (Cℓ+m )2 3 (ℓ!) Gn,m,ℓ = , (ℓ!)3 { (n+2m) 2 Hn,m,ℓ (n+m) (m) (n) [ (m) ]2 } , Hn,m = (−1)n Hℓ Hℓ − 2Hℓ Hℓ Hℓ + Hℓ Hℓ = (ℓ!)3 { (n+2m) 2 In,m,ℓ (n) (2m) (n) [ (m) ]2 } In,m = (−1)n Hℓ Hℓ − 2Hℓ Hℓ Hℓ + Hℓ Hℓ = . (ℓ!)3 Accordingly, the inequalities in (1.19), (1.20), and (1.21) are proved. The proof of  Theorem 1.7 is complete. 3. Remarks Remark 3.1. It is clear that the integral representation (1.10) is simpler and more significant than (1.3) for Cn . Remark 3.2. In the proof of [13, Theorem 4.1], the inequality (4.11) should be cn+ai +aj ≥ 0. n! m

12

F. QI, X.-T. SHI, AND F.-F. LIU

Consequently, the inequality (4.2) in [13, p. 248, Theorem 4.1] should be corrected as |cn+ai +aj |m ≥ 0. Remark 3.3. In recent years, the first author of this paper and his coauthors established integral representations, complete monotonicity, inequalities, closed expressions for the Lah numbers, tangent numbers, Bernoulli numbers of the first and second kinds, Stirling numbers of the first and second kind, Bernoulli polynomials, Euler numbers and polynomials. For detailed information, please refer to [4, 16, 17, 18, 19, 31, 38] and closely related references therein. Remark 3.4. This paper is a companion of the articles [12, 28, 29, 33]. References [1] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., Dordrecht and Boston, 1974. [2] A. M. Fink, Kolmogorov-Landau inequalities for monotone functions, J. Math. Anal. Appl. 90 (1982), 251–258; Available online at http://dx.doi.org/10.1016/0022-247X(82)90057-9. [3] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics—A Foundation for Computer Science, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. [4] B.-N. Guo and F. Qi, Some integral representations and properties of Lah numbers, J. Algebra Number Theory Acad. 4 (2014), no. 3, 77–87. [5] G. H. Hardy, J. E. Littlewood, and G. P´ olya, Inequalities, Cambridge University Press, Cambridge, 1934. [6] A.-Q. Liu, G.-F. Li, B.-N. Guo, and F. Qi, Monotonicity and logarithmic concavity of two functions involving exponential function, Internat. J. Math. Ed. Sci. Tech. 39 (2008), no. 5, 686–691; Available online at http://dx.doi.org/10.1080/00207390801986841. [7] A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of Majorization and its Applications, 2nd Ed., Springer Verlag, New York-Dordrecht-Heidelberg-London, 2011; Available online at http://dx.doi.org/10.1007/978-0-387-68276-1. [8] D. S. Mitrinovi´ c and J. E. Peˇ cari´ c, On some inequalities for monotone functions, Boll. Un. Mat. Ital. B (7) 5 (1991), no. 2, 407–416. [9] D. S. Mitrinovi´ c and J. E. Peˇ cari´ c, On two-place completely monotonic functions, Anzeiger ¨ Oster. Akad. Wiss. Math.-Natturwiss. Kl. 126 (1989), 85–88. [10] D. S. Mitrinovi´ c, J. E. Peˇ cari´ c, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht-Boston-London, 1993; Available online at http: //dx.doi.org/10.1007/978-94-017-1043-5. [11] J. E. Peˇ cari´ c, Remarks on some inequalities of A. M. Fink, J. Math. Anal. Appl. 104 (1984), no. 2, 428–431; Available online at http://dx.doi.org/10.1016/0022-247X(84)90006-4. [12] F. Qi, A logarithmically completely monotonic function involving the gamma function and originating from the Catalan numbers, ResearchGate Research, available online at http: //dx.doi.org/10.13140/RG.2.1.1401.2009. [13] F. Qi, An integral representation, complete monotonicity, and inequalities of Cauchy numbers of the second kind, J. Number Theory 144 (2014), 244–255; Available online at http://dx. doi.org/10.1016/j.jnt.2014.05.009. [14] F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010 (2010), Article ID 493058, 84 pages; Available online at http://dx.doi.org/10.1155/2010/493058. [15] F. Qi, Bounds for the ratio of two gamma functions: from Gautschi’s and Kershaw’s inequalities to complete monotonicity, Turkish J. Anal. Number Theory 2 (2014), no. 5, 152–164; Available online at http://dx.doi.org/10.12691/tjant-2-5-1. [16] F. Qi, Derivatives of tangent function and tangent numbers, Appl. Math. Comput. 268 (2015), 844–858; Available online at http://dx.doi.org/10.1016/j.amc.2015.06.123. [17] F. Qi, Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind, Filomat 28 (2014), no. 2, 319–327; Available online at http: //dx.doi.org/10.2298/FIL1402319O.

AN INTEGRAL REPRESENTATION OF THE CATALAN NUMBERS

13

[18] F. Qi, Integral representations and properties of Stirling numbers of the first kind, J. Number Theory 133 (2013), no. 7, 2307–2319; Available online at http://dx.doi.org/10.1016/j. jnt.2012.12.015. [19] F. Qi and R. J. Chapman, Two closed forms for the Bernoulli polynomials, J. Number Theory (2016), in press; Available online at http://arxiv.org/abs/1506.02137. [20] F. Qi and C.-P. Chen, A complete monotonicity property of the gamma function, J. Math. Anal. Appl. 296 (2004), 603–607; Available online at http://dx.doi.org/10.1016/j.jmaa. 2004.04.026. [21] F. Qi and B.-N. Guo, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 8, 63–72; Available online at http://rgmia.org/v7n1.php. [22] F. Qi, S. Guo, and B.-N. Guo, Complete monotonicity of some functions involving polygamma functions, J. Comput. Appl. Math. 233 (2010), no. 9, 2149–2160; Available online at http: //dx.doi.org/10.1016/j.cam.2009.09.044. [23] F. Qi and W.-H. Li, A logarithmically completely monotonic function involving the ratio of gamma functions, J. Appl. Anal. Comput. 5 (2015), no. 4, 626–634; Available online at http://dx.doi.org/10.11948/2015049. [24] F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions—From Wendel’s and related inequalities to logarithmically completely monotonic functions, Banach J. Math. Anal. 6 (2012), no. 2, 132–158; Available online at http://dx.doi.org/10.15352/bjma/1342210165. [25] F. Qi and Q.-M. Luo, Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to Elezovi´ c-Giordano-Peˇ cari´ c’s theorem, J. Inequal. Appl. 2013, 2013:542, 20 pages; Available online at http://dx.doi.org/10.1186/1029-242X-2013-542. [26] F. Qi, Q.-M. Luo, and B.-N. Guo, Complete monotonicity of a function involving the divided difference of digamma functions, Sci. China Math. 56 (2013), no. 11, 2315–2325; Available online at http://dx.doi.org/10.1007/s11425-012-4562-0. [27] F. Qi, Q.-M. Luo, and B.-N. Guo, The function (bx − ax )/x: Ratio’s properties, In: Analytic Number Theory, Approximation Theory, and Special Functions, G. V. Milovanovi´ c and M. Th. Rassias (Eds), Springer, 2014, pp. 485–494; Available online at http://dx.doi.org/10. 1007/978-1-4939-0258-3_16. [28] F. Qi, X.-T. Shi, and F.-F. Liu, An exponential representation for a function involving the gamma function and originating from the Catalan numbers, ResearchGate Research, available online at http://dx.doi.org/10.13140/RG.2.1.1086.4486. [29] F. Qi, X.-T. Shi, and F.-F. Liu, Several formulas for special values of the Bell polynomials of the second kind and applications, ResearchGate Technical Report, available online at http: //dx.doi.org/10.13140/RG.2.1.3230.1927. [30] F. Qi, C.-F. Wei, and B.-N. Guo, Complete monotonicity of a function involving the ratio of gamma functions and applications, Banach J. Math. Anal. 6 (2012), no. 1, 35–44; Available online at http://dx.doi.org/10.15352/bjma/1337014663. [31] F. Qi and X.-J. Zhang, An integral representation, some inequalities, and complete monotonicity of the Bernoulli numbers of the second kind, Bull. Korean Math. Soc. 52 (2015), no. 3, 987–998; Available online at http://dx.doi.org/10.4134/BKMS.2015.52.3.987. [32] R. L. Schilling, R. Song, and Z. Vondraˇ cek, Bernstein Functions—Theory and Applications, 2nd ed., de Gruyter Studies in Mathematics 37, Walter de Gruyter, Berlin, Germany, 2012; Available online at http://dx.doi.org/10.1515/9783110269338. [33] X.-T. Shi, F.-F. Liu, and F. Qi, An integral representation of the Catalan numbers, ResearchGate Research, available online at http://dx.doi.org/10.13140/RG.2.1.2273.6485. [34] R. Stanley and E. W. Weisstein, Catalan Number, From MathWorld–A Wolfram Web Resource; Available online at http://mathworld.wolfram.com/CatalanNumber.html. [35] H. van Haeringen, Completely monotonic and related functions, J. Math. Anal. Appl. 204 (1996), no. 2, 389–408; Available online at http://dx.doi.org/10.1006/jmaa.1996.0443. [36] H. van Haeringen, Inequalities for real powers of completely monotonic functions, J. Math. Anal. Appl. 210 (1997), no. 1, 102–113; Available online at http://dx.doi.org/10.1006/ jmaa.1997.5376. [37] I. Vardi, Computational Recreations in Mathematica, Addison-Wesley, Redwood City, CA, 1991. [38] C.-F. Wei and F. Qi, Several closed expressions for the Euler numbers, J. Inequal. Appl. 2015, 2015:219, 8 pages; Available online at http://dx.doi.org/10.1186/s13660-015-0738-9.

14

F. QI, X.-T. SHI, AND F.-F. LIU

[39] D. V. Widder, The Laplace Transform, Princeton Mathematical Series 6, Princeton University Press, Princeton, N. J., 1941. [40] Wikipedia, Catalan number, From The Free Encyclopedia; Available online at https://en. wikipedia.org/wiki/Catalan_number. [41] S.-Q. Zhang, B.-N. Guo, and F. Qi, A concise proof for properties of three functions involving the exponential function, Appl. Math. E-Notes 9 (2009), 177–183. (Qi) Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China; College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China E-mail address: [email protected], [email protected], [email protected] URL: https://qifeng618.wordpress.com (Shi) Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China E-mail address: [email protected], [email protected] (Liu) Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China E-mail address: [email protected]