INTEGRAL REPRESENTATIONS OF CATALAN NUMBERS AND THEIR APPLICATIONS

In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several power series whose coefficients involve the Catalan numbers.

The Catalan numbers C n for n ≥ 0 form a sequence of natural numbers that occur in various counting problems in combinatorial mathematics. The nth Catalan number can be expressed in terms of the central binomial coefficients 2n n by C n = 1 n + 1 2n n = (2n)! n!(n + 1)! . (1.1) The Catalan numbers C n were described in the 18th century by Leonhard Euler and are named after the Belgian mathematician Eugéne Charles Catalan. In 1988, it came to light that the Catalan numbers C n had been used in China by the Mongolian mathematician Ming Antu by 1730. See [18,19,20,22,23,24,25,26,64]. In recent years, the Catalan numbers C n has been analytically generalized and studied in [21,27,40,41,42,43,44,45,50,52,56,57,58,61,67,69,70] and the closely related references therein. For more information on the Catalan numbers C n , please refer to the monographs [10,15,59,63] and the closely related references therein.

Integral representations of the Catalan numbers
In this section, we recall integral representations of the Catalan numbers C n and their reciprocals 1 Cn and sketch their proofs as possible as we can.  in [62], it follows immediately that where (y) α−1 + = y α−1 , y > 0; 0, y < 0, the classical beta function B(z, w) can be defined by for (z), (w) > 0. Then the desired integral representation of C n is proved.
where erf(x) denotes the error function defined by Proof. We recite the proof in [33] as follows. This follows from applying the formula in [32, p. 29, Entries 1.14.39 and 1.14.40] to h(x) = e −x and the function f (x) in (2.4).
By similar arguments, Penson and Sixdeniers [33] also derived and an integral representation of the sequence B n C n , where B n is the Bell numbers [11,34,39,55] and Ei(y) is the exponential integral function which can be defined by and Proof. Now we sketch the proof in [7]. Let (2.10) Then I 0 = π 4 a 2 and I n (a) = a 2 n − 1 n + 2 I n−2 (a).
Substituting (1.1) into the above equations and making use of (2.10) result in (2.12) Further setting a = 2 leads to (2.8) and (2.9) immediately. (2.14) Proof. The sketch of the proof in [5] can be written as follows. For n ≥ 0, let By the substitution x = sin u for u ∈ 0, π 2 , we can deduce Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 7 April 2017 doi:10.20944/preprints201704.0040.v1 Considering the well-known fact that and using the expression (1.1) derive Accordingly, we acquire The integral representations (2.13) and (2.14) are thus proved.
The outline of the proof in [6]. It was stated in [14] that See also [36,p. 16,Eq. (2.18)]. Then it is not difficult to obtain On the other hand, using three irrational substitutions u 2 = 1 x 2 − 1, u 2 = 1 − x 2 , and u = 1−x 1+x to compute I n produces A new proof the formula (2.16). In [9, p. 325], the fourth formula reads that for 0 < µ ν < n + 1 and p, q = 0. Setting p = q = 1, µ = 3, and ν = 2 and replacing n by n + 1 find where we used in the last step the observation (2.20) The outline of the proof in [5]. Taking the substitution u = 1−x 1+x concludes Combining this for even n with (2.15), we derive the integral presentation (2.20) immediately.
The outline of the proof in [6]. By same argument as in the proof of Theorem 2.4 and by the third formula in (2.18), the integral representation (2.20) is verified once again.

Dana-Picard-Zeitoun-Qi's integral representations in 2012 and 2016.
In 2012, Dana-Picard and Zeitoun [8] deduced an integral representation for the Catalan numbers C n , which was corrected and developed by Qi [35] as the following integral representations. . For n ≥ 0 and a > 0, the Catalan numbers C n can be represented by and (2.22) Proof. We sketch the proof in [35]. Let a be a positive number. For n ≥ 0, define and J n = a n+1 π 1 + (−1) n n (2.25) The Catalan numbers C n can be expressed in terms of the beta function B(x, y) by Taking n = 2p in (2.24) and utilizing (2.26) lead to The first formula (2.21) thus follows. By similar argument to the deduction of (2.26), we can discover Employing this identity and setting n = 2p + 1 in ( The first formula in (2.22) is thus proved. The rest integral representations follow from mathematical techniques and changing variable of integration.  . For x ≥ 0, the Catalan function C x can be represented by Proof. This follows straightforwardly from applying the well-known formula in [65, (3.22)] to the logarithm of the Catalan function C x . for the Catalan numbers C n . Consequently, they derived an integral representation of the Catalan numbers C n .
Theorem 2.9 ([54, Theorem 1.4]). The Catalan numbers C n for n ≥ 0 can be represented by (2.28) Proof. The Catalan numbers C n can be generated by By virtue of the Cauchy integral formula in the theory of complex functions, we discover 1 4 . Therefore, it follows that Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 7 April 2017 doi:10.20944/preprints201704.0040.v1 Further using the substitution √ t = s yields the second integral representation in (2.28). The theorem is thus proved.

Qi's integral representations in 2017.
Theorem 2.10 ([38, Theorem 3.1 and Remark 6.6]). The Catalan numbers C n for n ≥ 0 can be represented by Proof. Using the substitution x = a sin s for s ∈ 0, π 2 and employing (2.17) for t = n ≥ 0 reveal for a > 0 and n ≥ 0. Differentiating with respect to a on both sides of (2.10) gives On the other hand, differentiating with respect to a on both sides of (2.31) results in Combining (2.32) with (2.33) and simplifying lead to for a > 0 and n ≥ 0. The first representation in (2.30) follows from combining The second integral representation in (2.30) follows immediately from combining (2.5) and (2.19). The desired proof is complete.
for k > 0 and n ∈ N was established.

The Catalan-Qi function and its integral representations
In 2015, Qi and his coauthors generalized in [53, Remark 1] and its formally published version [58,Eq. (9)] the Catalan numbers C n as the so-called Catalan-Qi function It is clear that .
When taking x = n ∈ {0} ∪ N, we call the quantities C(a, b; n) the Catalan-Qi numbers. It is easy to see that is called the rising factorial or the Pochhammer symbol. It is well known that the Wallis ratio is defined by Hence, it is easy to see that The Wallis ratio, or say, the ratio of two gamma functions, has been studied and applied by many mathematicians, see [12,36,37,46,47,48,49,51], for example, and plenty of literature therein. Now we are in a position to recall and to alternatively prove three integral representations of the Catalan-Qi function C(a, b; x) as follows.
Proof. This follows from combination of the definition (3.1) and the integral formula in [65, p. 67] for the ratio of two gamma functions Γ(z + a) and Γ(z + b).  and (3.6) An alternative proof. Making use of the last formula in (2.5) and the definition (3.1), we can rewritten the Catalan-Qi function C(a, b; x) as respectively. The proof of Theorem 3.2 is thus complete.

Discussing various integral representations
In this section, we will discuss various integral representations recalled and proved above.

4.2.
Discussing (2.6). By (2.35) and Γ(n + 1) = n!, we obtain Combining this with (2.6) and simplifying give Hence, we guess that Actually, this can be derived from by integration by part and the definition (2.7). In a word, we proved the integral representation (2.6) alternatively.

Discussing Theorems 2.3 and 2.4.
By the substitution x = 2t, the integral representations (2.8) and (2.9) reduce to (2.13) and (2.14). This can also be showed by letting a = 1 in (2.8) and (2.9). Consequently, the integral representations (2.8) and (2.9) are respectively equivalent to (2.13) and (2.14). By the substitution x = √ t in (2.13) and by the first definition in (2.5), we obtain Accordingly, the integral representation (2.13) is a special case of the integral representation (3.5) and is equivalent to (2.1).
Similarly, by the substitution x = √ t in (2.14) and by the first definition in (2.5), we acquire 1 (4.1) This implies that the integral representations (2.9) and (2.14) for reciprocals of the Catalan numbers C n can be alternatively verified by using (2.35) and (2.5) in sequence as follows: for a > 0 and n ≥ 0.
Thus, the integral representations in (2.28) are proved alternatively. When changing the variable of integration by t = u 2 in the last representation in (2.28), we can recover the integral representation (2.16).

Discussing (2.30). The first integral in (2.30) can be computed as
Then from (2.26) it follows that √ π , and the recurrence relation Γ(x + 1) = xΓ(x), it is easy to see that which is different from the one in (4.1). Indeed, the Catalan numbers C n and their reciprocals 1 Cn can also be represented in terms of the beta functions B n + −

Applications of integral representations
Most of the above integral representations can be applied to discover properties of the Catalan numbers C n . Now we recall some known applications of several integral representations of the Catalan numbers C n . 5.1. The integral representation (2.1) was applied in the proof of [42, Theorem 5.1] to discover the identity This identity generalizes Recall from [66, p. 163, Definition 14a] that a completely monotonic sequence {a n } n≥0 is minimal if it ceases to be completely monotonic when a 0 is decreased. Let λ = (λ 1 , λ 2 , . . . , λ n ) ∈ R n and µ = (µ 1 , µ 2 , . . . , µ n ) ∈ R n . A sequence λ is said to be majorized by µ (in symbols λ µ) if 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, For more information on the theory of majorization and its applications, please refer to monographs [13,29] and the closely related references therein. Applying the integral representation (2.28), we can obtain properties and inequalities of the Catalan numbers C n . Some of them can be recited as follows.  ≥ 1 and a 0 , a 1 , . . . , a m be non-negative integers, then C a0 4 a0 where |e kj | m denotes a determinant of order m with elements e kj .
where C is defined by (5.5). Consequently, (1) the infinite sequence {C n } n≥0 is logarithmically convex, (2) the inequality C n +k ≤ C k +n C n−k (5.7) is valid for ≥ 0 and n > k > 0.
Theorem 5.5 ([54, Theorem 1.7]). If ≥ 0, n ≥ k ≥ m, k ≥ n−k, and m ≥ n−m, then For n, m ∈ N and ≥ 0, let where C is defined by (5.5). Then By virtue of the integral representation (3.5), we obtain asymptotic expansions and complete monotonicity related to the Catalan-Qi function.
is the falling factorial. Consequently, the function , where x denotes the floor function whose value is the largest integer less than or equal to x.

Power series whose coefficients involve Catalan numbers
In this section, we recall some results on sums of power series whose coefficients involve the Catalan numbers C n or the Catalan-Qi numbers C ( a, b; n). [16] proved the following theorem.

In 2012, Koshy and Gao
Proof. We reformulate the proof by Koshy and Gao in [16] as follows. Denote x n C n . Then Since n+2 Cn = 4n+2 Cn+1 , by the recurrence relation, this yields Using (6.4), this can be rewritten as Using (6.4) again gives Consequently, where α 1 is a constant. For 0 < x < 4, we have where α 2 is also a constant. Therefore, we have where α = 2α 2 − α 1 . Since f (0) = 1 = f (0), we have α = 0. Thus, the desired result for 0 < x < 4 is proved. For −4 < x < 0, by similar argument to the above, we acquire The desired result is thus proved.

In 2014, Beckwith and Harbor [4] proposed a problem: show that
In 2016, Abel [1] answered this problem by proving a general result below.
Proof. We slightly modify the proof in [1] as follows. Using the beta integral Direct calculation of the integral yields the result (6.5).
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 7 April 2017 doi:10.20944/preprints201704.0040.v1 6.3. The editorial comment in [1] listed the formulas The editorial comment in [1] also pointed out that the result (6.1) had existed in [16], that the sum can be found on the website http://planetmath.org/, and that the problem by Beckwith and Harbor [4] can be solved easily from which are special cases of the general formula in [17, p. 452, Theorem] below. Proof of (6.7). Making use of the familiar Gregory series and setting t = x √ 1−x 2 yields arctan t = arcsin x and .
From (6.7), Lehmer [17] also derived 6.4. In 2016, motivated by the above-mentioned problem posed by Beckwith and Harbor [4], Amdeberhan and his four coauthors [3] also proposed a general problem: find a closed-form formula for the series in (6.2). They obtained the sum ∞ n=0 z n C n = 2 F 1 1, 2; by several methods, where 2 F 1 is the classical hypergeometric function which is a special case of the generalized hypergeometric series x(x + 1) · · · (x + n − 1), n ≥ 1; 1, n = 0.
We observe that the formulas (6.5) and (6.6) are the same one, that the sums (6.1) and (6.8) are the same one, and that, since for 0 ≤ x < 4, the four sums (6.5) to (6.8) are essentially the same one.
Furthermore, similarly integrating gives The proof of the formula (6.7) is complete.

Remarks
Finally we list several remarks on closely related results.
Remark 9.3. Letting a = 1 2 and b = 2 in (7.6) and comparing with (2.29) leads to This can also be deduced from the formula