Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine functions and applications

di Abstract. In the paper, by means of the Fa`a di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a speciﬁc sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the ﬁrst kind, the author establishes Maclaurin’s series expansions for real powers of the inverse cosine function and the inverse hyperbolic cosine function. By applying diﬀerent series expansions for the square of the inverse cosine function, the author not only ﬁnds inﬁnite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coeﬃcients.

where s(n, k) denotes the first kind Stirling numbers which can be analytically computed by For m ∈ N and |x| < ∞, the function arcsinh x x m , whose value at x = 0 is defined to be 1, has the nice Maclaurin series expansion where s(n, k) stands for the first kind Stirling numbers generated by (1.5). where s(n, m) denotes the Stirling numbers of the first kind which can be analytically generated by In [11,12], the series expansion (1.2) was applied to derive closed-form formulas for special values of the Bell polynomials of the second kind, asked in [19] when studying Grothendieck's inequality and completely correlation preserving functions, and was applied to establish series representations of the generalized logsine function, considered in [9,15].
For k ≥ 2 and |x| < 1, we have The series expansions (1.6), (1.7), (1.8), and (1.9) in Theorem 1.3 are Taylor's series expansions at the point x = 1 of even powers of the inverse cosine function arccos x and the inverse hyperbolic cosine function arccosh x in terms of the Stirling numbers of the first kind s(n, m).
What are Maclaurin's series expansions at the point x = 0 for (arccos x) α and (arccosh x) α with α ∈ R? In Section 3 of this paper, we will answer this interesting and significant question. Besides this, we also (1) establish explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence 1, 0, 1, 0, 9, 0, 225, . . . , see Theorem 2.1 below; (2) apply different series expansions of (arccos x) 2 , including Taylor's series expansion (1.6) in Theorem 1.3, to find infinite series representations of π and π 2 respectively, see Theorem 4.1 below; (3) apply different series expansions of (arccos x) 2 to derive two combinatorial identities involving central binomial coefficients, see The Faà di Bruno formula [7,Theorem 11.4] and [8, p. 139, Theorem C] can be described in terms of To establish Maclaurin's series expansions at the point x = 0 for real powers of the inverse cosine function arccos x and the inverse hyperbolic cosine function arccosh x, we need the following explicit formulas for special values of the Bell polynomials of the second kind with respect to the sequence 1, 0, 1, 0, 9, 0, 225, . . . , 0, [(2r − 1)!!] 2 , 0, . . . .
Proof. In [10, p. 60, 1.641], there is the series expansion The series expansion (2.4) means that At the end of [8, p. 133], there is the formula for k ≥ 0. Making use of the formula (2.6) yields , employing the values in (2.5), and utilizing the series expansion (1.2) in Theorem 1.1 give where the falling factorial ⟨z⟩ k of z ∈ C is defined by Further employing the identity where ⟨α⟩ r for α ∈ R and r ∈ N stands for the falling factorials defined by (2.8) and s(n, m) for n ≥ m ≥ 0 denotes the Stirling numbers of the first kind generated in (1.5).
Proof. Let u = u(x) = arccos x. It is clear that u = u(x) = arccos x → π 2 as x → 0. By means of the Faà di Bruno formula (2.1) and the values in (2.5), we obtain When k = 2r and r ≥ 2, it follows that where we used the identity (2.9) and Theorem 2.1.
It is easy to see that When k = 2r − 1 for r ≥ 2, it follows that where we used the identity (2.9) and Theorem 2.1.
It is easy to see that Proof. Setting α = 2 in the series expansion (3.1) in Theorem 3.1 arrives at 2 arccos x π Further using the identities 2k ℓ=0 (ℓ + 1)s(2k The series expansion (3.6) follows. The proof of Corollary 3.1 is complete. □ Corollary 3.2. For k ∈ N and |x| < 1, we have and where the falling factorials ⟨2k⟩ r for k, r ∈ N are defined by (2.8).

Infinite series representations related to Pi and its square
By means of comparing the series expansion (1.6) in Theorem 1.3 with the series expansion (3.6) in Corollary 3.1, we can find the following infinite series representations of π and π 2 .  Proof. Maclaurin's series expansion (3.6) can be reformulated as The series expansion (1.6) in Theorem 1.3 can be rearranged as Accordingly, we obtain Comparing the series expansion (4.4) with the series expansion (4.3) produces By taking the special value x = 1 2 on both sides of (3.1) in Theorem 3.1, we can obtain the following interesting series representation.
Proof. This follows from taking the special value x = 1 2 on both sides of (3.1) in Theorem 3.1 and interchanging the orders of sums. The proof of Theorem 4.2 is complete. □ Corollary 4.1. The circular constant π can be represented as Proof. From taking α = 1 on both sides of (4.6) in Theorem 4.2 and simplifying, we doscover Further making use of the identity (3.8) and simplifying, we conclude the series representation (4.7). The proof of Corollary 4.1 is complete. □ Proof. The series representation (4.9) comes from taking α = 2 on both sides of (4.6) in Theorem 4.2 and simplifying give Further using the formula s(r, 1) = (−1) r−1 (r − 1)! for r ∈ N, employing the identities (3.7) and (3.8), and simplifying, we acquire Substituting (4.7) in Corollary 4.1 into (4.9) reveals The series representation (4.8) is thus obtained. The proof of Corollary 4.2 is complete. □

Two combinatorial identities
Squaring on both sides of the equation (2.4) and comparing with Maclaurin's series expansion (3.6) in Corollary 3.1, as well as rewriting the equation (4.5), we can derive the following two combinatorial identities. Proof. From the series expansion (2.4) and Cauchy's product, we conclude that Comparing this with (4.3), equating coefficients of the factors x 2r , and simplifying reveal the combinatorial identity (5.1). The equation (4.5) can be simplified as The combinatorial identity (5.2) is thus proved. The proof of Theorem 5.1 is complete. □
Remark 6.3. Maclaurin's series expansion (3.9) can be reformulated as The series expansions (4.3) and (6.1) are more beautiful and concise in form.