SEVERAL FORMULAS FOR SPECIAL VALUES OF THE BELL POLYNOMIALS OF THE SECOND KIND AND APPLICATIONS

In the paper, the authors establish several explicit formulas for special values of the Bell polynomials of the second kind, connect these formulas with the Bessel polynomials, and apply these formulas to give new expressions for the Catalan numbers and to compute arbitrary higher order derivatives of elementary functions such as the since, cosine, exponential, logarithm, arcsine, and arccosine of the square root for the variable.


Notation and main results
In combinatorial analysis, the Bell polynomials of the second kind, also known as the partial Bell polynomials, denoted by B n,k (x 1 , x 2 , . . . , x n−k+1 ), can be defined by B n,k (x 1 , x 2 , . . . , x n−k+1 ) = 1≤i≤n, i∈{0}∪N for n ≥ k ≥ 0. The well-known Faà di Bruno formula can be described in terms of the Bell polynomials of the second kind B n,k (x 1 , x 2 , . . . , x n−k+1 ) by d n d x n f • g(x) = n k=0 f (k) (g(x))B n,k g (x), g (x), . . . , g (n−k+1) (x) , (1.1) The first aim of this paper is to discover an explicit formula for special values In recent years, several explicit formulas of special values for the Bell polynomials of the second kind B n,k (x 1 , x 2 , . . . , x n−k+1 ) were discovered, recovered, and applied in [4,6,19,22,30,37] and references cited therein.
For more information on the Bessel polynomials y n (x), please refer to the web sites [33,34,38]. The second aim is to simplify the right hand side of the identity (1.3) in Theorem 1.1 and, as a consequence, to find a connection between special values of the quantity (1.2) and coefficients b n,k , defined in (1.4), of the Bessel polynomials y n (x).
Consequently, coefficients b n,k of the Bessel polynomials y n (x) and special values of the quantity (1.2) satisfy In combinatorial number theory, the Catalan numbers C n 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?" The first twelve Catalan numbers C n for 0 ≤ n ≤ 11 are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786. Let us recall some conclusions in [3,8,35,36] as follows. Explicit formulas of C n for n ≥ 0 include is the classical Euler gamma function and is the generalized hypergeometric series defined for positive integers p, q ∈ N, for complex numbers a i ∈ C and b i ∈ C \ {0, −1, −2, . . . }, and in terms of the rising factorials The asymptotic form for the Catalan function Motivated by the sixth expression in (1.7) and by virtue of an integral representation of the gamma function ln Γ(x), the authors represented in [32,Theorem 1] the Catalan function C x as Hereafter, the above integral representation was further deeply cultivated in [15,29]. For more detailed information on the Catalan numbers C n , please refer to the monographs and websites [2,3,35,39] and references cited therein. The third aim of this paper is to apply the formulas (1.3) and (1.5) in Theorems 1.1 and 1.2 to express the Catalan numbers as new forms below. Theorem 1.3. For n ≥ 0, the Catalan numbers can be expressed by Finally, the fourth aim of this paper is to apply the formula (1.5) in Theorem 1.2 to compute arbitrary higher order derivatives of several elementary functions of the form f ( √ a + bx ) for a, b ∈ R and b = 0. As examples, we obtain the following results.
Theorem 1.4. Let g(x) = √ a + bx for a, b ∈ R and b = 0 and let n ∈ N. Then the Bell polynomials of the second kind B n,k satisfy (1.10) Consequently, for n ≥ 0, we have Consequently, for n ≥ 1, we have

Proofs
Differentiating m ≥ k times on both sides of the above equation reveals that Further letting t → 0 yields where the formula B n,k abx1, ab 2 x2, . . . , ab n−k+1 x n−k+1 = a k b n B n,k (x1, x2, . . . , x n−k+1 ) (2.3) for complex numbers a and b, which was listed in [2, p. 135], was employed above. Proof. [Proof of Theorem 1.2] In [33], it was mentioned that On the other hand, as done in (2.2), it follows that Since the sequence of the functions (1 − 2x) k/2−n is linearly independent, we have that is, the identity (1.5) follows immediately. The equality (1.6) comes from directly verifying b n,k = T (n, n − k). The proof of Theorem 1.2 is complete. Proof. [Proof of Theorem 1.3] The Catalan numbers C n can be generated [35,39] by Hence, making use of the equations (2.2) and (2.4) and employing Theorem 1.1, it follows that  sin As did just now, considering cos (n) x = cos x + n π 2 leads to the formula for the derivative d n d x n cos √ x in Theorem 1.4. Similarly, we acquire Furthermore, we see that In the proof of [30, Theorem 3.1, pp. 601-602], it was derived that 1, 0, . . . , 0), n ≥ 0.
In [6,Theorem 4.1], it was established that the Bell polynomials of the second kind B n,k for 0 ≤ k ≤ n satisfy for n ≥ 1. The formula (1.11) is proved.
By the Faà di Bruno formula (1.1), the formula (1.10) applied to a = 0 and b = 1, and the formula (1.11), we arrive at for n ≥ 1. The proof of Theorem 1.4 is complete.

Remarks
Finally, we give several remarks on the recovery of the third formula in (1.7), on the formulas (1.11) and (2.8), and on the derivatives of some elementary functions.
Remark 3.1. Let u = u(z) and v = v(z) = 0 be differentiable functions. In [1, p. 40], the formula for the kth derivative of the ratio u(z) v(z) was listed. In [ where the matrices under the conventions that v (0) (z) = v(z) and that p q = 0 and v (p−q) (z) ≡ 0 for p < q. Applying the formula (3.1) or (3.2) to the function 1− Consequently, we obtain This implies that As a result, by virtue of (2.7), we acquire which recovers the third formula in (1.7) for the Catalan numbers C n for n ≥ 0.
Remark 3.2. On 14 September 2015, when the original version of this paper was finalized, we found the papers [13,14,17] which are related to the generating function (2.7) of the Catalan numbers C n and its derivatives.  [19]. By virtue of conclusions obtained in [5,7,41], some nice closed formulas for higher order derivatives of the tangent, cotangent, secant, cosecant, hyperbolic tangent, hyperbolic cotangent, hyperbolic secant, and hyperbolic cosecant functions were found in [40]. Hence, we can regard this paper as a companion of the papers [19,30,40] and closely related reference therein.

Appendix
For completeness and accuracy, we now directly and alternatively verify the formulas (2.5) and (2.6) as follows.
In [10][11][12] and closely-related references therein, the composita Y ∆ (n, k, x) of the function Y (x, z) = y(x + z) − y(x) was introduced by where y (i) is the ith derivative of the function y(x), S n is the set of compositions of n, and π k is the composition of n with k parts such that k =1 λ = n. In those papers, it was proved that the composita Y ∆ (n, k, x) can be generated by [Y (x, z)] k = [y(x + z) − y(x)] k = ∞ n=k Y ∆ (n, k, x)z n (4.1) and that the Bell polynomials B n,k and the composita Y ∆ (n, k, x) have the relation B n,k y (x), y (x), . . . , y (n−k+1) (x) = n! k! Y ∆ (n, k, x).

(4.2)
It is straightforward that Hence, by (4.1), the composita F ∆ (n, k, x) of the function F (x, z) = f (x + z) − f (x) is equal to Therefore, by virtue of the relation (4.2), the Bell polynomial of the second kind B n,k for the function f ( which can be reformulated as (2.5) and (2.6).