Power Series Expansions of Real Powers of Inverse Cosine and Sine Functions, Closed-Form Formulas of Partial Bell Polynomials at Specific Arguments, and Series Representations of Real Powers of Circular Constant ^†

Abstract

In this paper, by means of the Faà di Bruno formula, with the help of explicit formulas for partial Bell polynomials at specific arguments of two specific sequences generated by derivatives at the origin of the inverse sine and inverse cosine functions, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes power series expansions for real powers of the inverse cosine (sine) functions and the inverse hyperbolic cosine (sine) functions. By comparing different series expansions for the square of the inverse cosine function and for the positive integer power of the inverse sine function, the author not only finds infinite series representations of the circular constant $\pi$ and its real powers, but also derives several combinatorial identities involving central binomial coefficients and the Stirling numbers of the first kind.

1. Preliminaries and Motivations

In this paper, basing on conventions in community of mathematics, we adopt the notations

$$\begin{array}{ccc}\begin{array}{cc}\hfill \mathbb{N}& =\{1,2,\dots \},\hfill \\ \hfill \mathbb{Z}& =\{0,\pm 1,\pm 2,\dots \},\hfill \end{array}\hfill & \begin{array}{cc}\hfill {\mathbb{N}}_{-}& =\{-1,-2,\dots \},\hfill \\ \hfill \mathbb{R}& =(-\infty ,\infty ),\hfill \end{array}\hfill & \begin{array}{cc}\hfill {\mathbb{N}}_0& =\{0,1,2,\dots \},\hfill \\ \hfill \mathbb{C}& =\left\{x+\mathrm{i}y:x,y\in \mathbb{R},\mathrm{i}=\sqrt{-1}\right\}.\hfill \end{array}\hfill \end{array}$$

The classical Euler gamma function $\Gamma \left(w\right)$ can be defined [1] [Chapter 3] by

$$\Gamma \left(w\right)=\underset{m\to \infty }{\lim }{\displaystyle \frac{m!m^w}{{\prod }_{k=0}^m(w+k)}},w\in \mathbb{C}\setminus \{0,-1,-2,\dots \}.$$

The partial Bell polynomials, also known as the Bell polynomials of the second kind, are denoted and defined in [2] [Definition 11.2] and [3] [p. 134, Theorem A] by

$$B_{m,k}(x_1,x_2,\dots ,x_{m-k+1})=\sum _{\begin{array}{c}1\le j\le m-k+1,{\ell }_j\in \left\{0\right\}\cup \mathbb{N}\\ {\sum }_{j=1}^{m-k+1}j{\ell }_j=m,{\sum }_{j=1}^{m-k+1}{\ell }_j=k\end{array}}{\displaystyle \frac{m!}{{\prod }_{j=1}^{m-k+1}{\ell }_j!}}\prod _{j=1}^{m-k+1}{\left({\displaystyle \frac{x_j}{j!}}\right)}^{{\ell }_j}.$$

The Faà di Bruno formula can be represented via partial Bell polynomials $B_{m,k}$ by

$${\displaystyle \frac{{\mathrm{d}}^m}{\mathrm{d}{z}^m}}F\circ H\left(z\right)=\sum _{k=1}^m{F}^{\left(k\right)}\left(H\left(z\right)\right)B_{m,k}\left(H^{\prime }\left(z\right),H^{″}\left(z\right),\dots ,H^{(m-k+1)}\left(z\right)\right)$$

(1)

for $m\in \mathbb{N}$. See [2] [Theorem 11.4] and [3] [p. 139, Theorem C].

The modified Bessel function of the first kind $I_{\tau }\left(w\right)$ can be represented [4] [p. 375, 9.6.10] by

$$I_{\tau }\left(w\right)=\sum _{m=0}^{\infty }{\displaystyle \frac{1}{m!\Gamma (\tau +m+1)}}{\left({\displaystyle \frac{w}{2}}\right)}^{2m+\tau },w\in \mathbb{C}.$$

The rising factorial ${\left(w\right)}_m$, also known as the Pochhammer symbol or shifted factorial, of a complex number $w\in \mathbb{C}$ is defined [5] [p. 7497] by

$${\left(w\right)}_m=\prod _{k=0}^{m-1}(w+k)=\left\{\begin{array}{cc}w(w+1)\cdots (w+m-1),\hfill & m\in \mathbb{N};\hfill \\ 1,\hfill & m=0.\hfill \end{array}\right.$$

(2)

The falling factorial of a complex number $w\in \mathbb{C}$ is defined [5] [p. 7497] by

$${\langle w\rangle }_m=\prod _{\ell =0}^{m-1}(w-\ell )=\left\{\begin{array}{cc}w(w-1)\cdots (w-m+1),\hfill & m\in \mathbb{N};\hfill \\ 1,\hfill & m=0.\hfill \end{array}\right.$$

(3)

The Stirling numbers of the first kind $s(n,k)$ for $n\ge k\in {\mathbb{N}}_0$ can be analytically generated [1] [p. 20, (1.30)] by

$${\displaystyle \frac{{\left[\ln (1+t)\right]}^k}{k!}}=\sum _{n=k}^{\infty }s(n,k){\displaystyle \frac{t^n}{n!}},\left|t\right|<1.$$

(4)

Equation (4) can be rearranged as a power series expansion:

$${\left[{\displaystyle \frac{\ln (1+t)}{t}}\right]}^k=\sum _{n=0}^{\infty }{\displaystyle \frac{s(n+k,k)}{\left(\genfrac{}{}{0pt}{}{n+k}{k}\right)}}{\displaystyle \frac{t^n}{n!}}$$

for $\left|t\right|<1$ and $k\in {\mathbb{N}}_0$.

The Stirling numbers of the second kind $S(n,k)$ for $n\ge k\in {\mathbb{N}}_0$ can be generated [6] [pp. 131–132] by

$${\displaystyle \frac{{({\mathrm{e}}^t-1)}^k}{k!}}=\sum _{n=k}^{\infty }S(n,k){\displaystyle \frac{t^n}{n!}}.$$

(5)

Equation (5) can be rewritten as a power series expansion:

$${\left({\displaystyle \frac{{\mathrm{e}}^t-1}{t}}\right)}^k=\sum _{n=0}^{\infty }{\displaystyle \frac{S(n+k,k)}{\left(\genfrac{}{}{0pt}{}{n+k}{k}\right)}}{\displaystyle \frac{t^n}{n!}},k\in {\mathbb{N}}_0.$$

(6)

In [7] [p. 377, (3.5)] and [8] [pp. 109–110, Lemma 1], it was obtained that

$$I_{\mu }\left(t\right)I_{\tau }\left(t\right)={\displaystyle \frac{1}{\Gamma (\mu +1)\Gamma (\tau +1)}}\sum _{m=0}^{\infty }{\displaystyle \frac{{(\mu +\tau +m+1)}_m}{m!{(\mu +1)}_m{(\tau +1)}_m}}{\left({\displaystyle \frac{t}{2}}\right)}^{2m+\mu +\tau }.$$

In [9] [p. 310], we find the power series expansion

$${\left[I_{\tau }\left(w\right)\right]}^2=\sum _{m=0}^{\infty }{\displaystyle \frac{1}{{[\Gamma (\tau +m+1)]}^2}}\left({\displaystyle \genfrac{}{}{0pt}{}{2m+2\tau }{m}}\right){\left({\displaystyle \frac{w}{2}}\right)}^{2m+2\tau },$$

where the extended binomial coefficient $\left({\displaystyle \genfrac{}{}{0pt}{}{z}{w}}\right)$ is defined [10] by

$$\left({\displaystyle \genfrac{}{}{0pt}{}{z}{w}}\right)=\{\begin{array}{ll}{\displaystyle \frac{\Gamma (z+1)}{\Gamma (w+1)\Gamma (z-w+1)}},& z\notin {\mathbb{N}}_{-},w,z-w\notin {\mathbb{N}}_{-};\\ 0,& z\notin {\mathbb{N}}_{-},w\in {\mathbb{N}}_{-}\mathrm{or}z-w\in {\mathbb{N}}_{-};\\ {\displaystyle \frac{{\langle z\rangle }_w}{w!}},& z\in {\mathbb{N}}_{-},w\in {\mathbb{N}}_0;\\ {\displaystyle \frac{{\langle z\rangle }_{z-w}}{(z-w)!}},& z,w\in {\mathbb{N}}_{-},z-w\in {\mathbb{N}}_0;\\ 0,& z,w\in {\mathbb{N}}_{-},z-w\in {\mathbb{N}}_{-};\\ \infty ,& z\in {\mathbb{N}}_{-},w\notin \mathbb{Z}.\end{array}$$

In general, the series expansions of the functions ${\left[I_{\nu }\left(w\right)\right]}^r$ for $\nu \in \mathbb{C}\setminus {\mathbb{N}}_{-}$ and $r,w\in \mathbb{C}$ have been surveyed and investigated in [9,11,12,13,14]. Because the products of the (modified) Bessel functions of the first kind $I_{\nu }\left(w\right)$ has ever appeared in the theory of statistical mechanics and plasma physics (see [15,16]), many mathematicians and physicists have been attracted to the investigation of the series expansions of the functions ${\left[I_{\nu }\left(w\right)\right]}^r$.

In the articles [5,17,18,19,20,21,22,23], power series expansions of the power functions

$$\begin{array}{ccccccc}\hfill {\sin }^{m}w,& \hfill & \hfill {\cos }^{m}w,& \hfill & \hfill {\tan }^{m}w,& \hfill & \hfill {\cot }^{m}w,\\ \hfill {\sec }^{m}w,& \hfill & \hfill {\csc }^{m}w,& \hfill & \hfill {\left(\arctan w\right)}^m,& \hfill & \hfill {\left(\mathrm{arctanh}w\right)}^m,\\ \hfill {\left({\displaystyle \frac{\arcsin w}{w}}\right)}^m,& \hfill & \hfill {\displaystyle \frac{{\left(\arcsin w\right)}^m}{\sqrt{1-w^2}}},& \hfill & \hfill {\left({\displaystyle \frac{\mathrm{arcsinh}w}{w}}\right)}^m,& \hfill & \hfill {\displaystyle \frac{{\left(\mathrm{arcsinh}w\right)}^m}{\sqrt{1+w^2}}}\end{array}$$

for $m\ge 2$ and their history have been reviewed, established, discussed, and applied. Now, we recite several nice series expansions as follows.

Theorem 1 

([19] [Theorem 1] and [24] [Section 6]). For $m\in \mathbb{N}$ and $\left|t\right|<1$, assume that the value of the function ${\left({\displaystyle \frac{\arcsin t}{t}}\right)}^m$ at $t=0$ is 1. Then,

$${\left({\displaystyle \frac{\arcsin t}{t}}\right)}^m=1+\sum _{n=1}^{\infty }{(-1)}^n{\displaystyle \frac{Q(m,2n)}{\left(\genfrac{}{}{0pt}{}{m+2n}{m}\right)}}{\displaystyle \frac{{\left(2t\right)}^{2n}}{\left(2n\right)!}},$$

(7)

where

$$Q(m,n)=\sum _{k=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{m+k-1}{m-1}}\right)s(m+n-1,m+k-1){\left({\displaystyle \frac{m+n-2}{2}}\right)}^k$$

(8)

for $m\in \mathbb{N}$ and $n\ge 2$, the Stirling numbers of the first kind $s(n,k)$ can be analytically computed by

$$\left|s(n+1,k+1)\right|=n!\sum _{{\ell }_1=k}^n{\displaystyle \frac{1}{{\ell }_1}}\sum _{{\ell }_2=k-1}^{{\ell }_1-1}{\displaystyle \frac{1}{{\ell }_2}}\cdots \sum _{{\ell }_{k-1}=2}^{{\ell }_{k-2}-1}{\displaystyle \frac{1}{{\ell }_{k-1}}}\sum _{{\ell }_k=1}^{{\ell }_{k-1}-1}{\displaystyle \frac{1}{{\ell }_k}},n\ge k\in \mathbb{N}.$$

(9)

Theorem 2 

([19] [Corollary 2] and [24] [Section 6]). For $m\in \mathbb{N}$ and $\left|t\right|<\infty$, assume that the value of the function ${\left({\displaystyle \frac{\mathrm{arcsinh}t}{t}}\right)}^m$ at $t=0$ is 1. Then,

$${\left({\displaystyle \frac{\mathrm{arcsinh}t}{t}}\right)}^m=1+\sum _{n=1}^{\infty }{\displaystyle \frac{Q(m,2n)}{\left(\genfrac{}{}{0pt}{}{m+2n}{m}\right)}}{\displaystyle \frac{{\left(2t\right)}^{2n}}{\left(2n\right)!}},$$

(10)

where $Q(m,2n)$ is defined by (8).

In [5,19], the series expansion (7) was applied to the following tasks:

• Derive closed-form expressions for partial Bell polynomials at specific arguments, where these closed-form formulas were asked for in [25];
• Establish series representations of the generalized logsine function, where these series representations were considered in [26,27].

Formula (9) is a reformulation of [28] [Corollary 2.3]. The series expansions (7) and (10) in Theorems 1 and 2 were also recovered in [24] [Section 6].

On the website https://math.stackexchange.com/q/3209345 (accessed on 1 May 2019), the author listed the first five concrete expressions of the series expansion (7) for $m=1,2,3,4,5$.

Theorem 3 

([24] [Theorem 3.1]). For $k\in \mathbb{N}$ and $\left|t\right|<1$, we have

$${\left[{\displaystyle \frac{{\left(\arccos t\right)}^2}{2(1-t)}}\right]}^k=1+\sum _{n=1}^{\infty }{\displaystyle \frac{Q(2k,2n)}{(2n-1)!!\left(\genfrac{}{}{0pt}{}{2n+2k}{2k}\right)}}{\displaystyle \frac{{(t-1)}^n}{n!}}$$

(11)

and

$${\left[{\displaystyle \frac{{\left(\mathrm{arccosh}t\right)}^2}{2(1-t)}}\right]}^k={(-1)}^k\left[1+\sum _{n=1}^{\infty }{\displaystyle \frac{Q(2k,2n)}{(2n-1)!!\left(\genfrac{}{}{0pt}{}{2n+2k}{2k}\right)}}{\displaystyle \frac{{(t-1)}^n}{n!}}\right],$$

(12)

where $Q(2k,2n)$ is defined by (8).

The series expansions (11) and (12) in Theorem 3 can be regarded as the Taylor series expansions at $t=1$ of even power functions ${\left(\arccos t\right)}^{2k}$ and ${\left(\mathrm{arccosh}t\right)}^{2k}$ in terms of the quantity $Q(m,n)$.

What are the power series expansions around the point $t=0$ for the power functions

$${\left(\arccos t\right)}^{\alpha },{\left(\mathrm{arccosh}t\right)}^{2m},{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\alpha },{\left({\displaystyle \frac{\mathrm{arcsinh}t}{t}}\right)}^{2m}$$

with $\alpha \in \mathbb{R}$ and $m\in \mathbb{N}$? In Section 3 and Section 4 of this paper, we will answer this interesting and significant question. Furthermore, we will also perform the following works:

• Establish closed-form formulas for partial Bell polynomials $B_{m,k}$ at specific arguments of the sequences

  $$1,0,1,0,9,0,225,\dots \mathrm{and}0,{\displaystyle \frac{1}{3}},0,{\displaystyle \frac{9}{5}},0,{\displaystyle \frac{225}{7}},0,\dots ;$$

  see Theorems 4 and 5.
• Apply two different series expansions of the square function ${\left(\arccos t\right)}^2$, including the Taylor series expansion (11) in Theorem 3 for $k=1$, and the power series expansion (43) of the function ${\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\alpha }$ for $\alpha \in \mathbb{R}$ in Theorem 7 to find infinite series representations of $\pi$, ${\pi }^2$, and ${\pi }^{\alpha }$ for $\alpha \in \mathbb{R}$, respectively; see Theorems 9 and 11.
• Apply two different series expansions of ${\left(\arccos t\right)}^2$ and compare the series expansion (7) in Theorem 1 with the series expansion (43) in Theorem 7 to derive two combinatorial identities involving central binomial coefficients; see Theorems 12 and 13.

2. Closed-Form Formulas for Partial Bell Polynomials

To establish power series expansions around the point $t=0$ for real powers of the functions $\arccos t$, $\mathrm{arccosh}t$, $\arcsin t$, and $\mathrm{arcsinh}t$, we need the following closed-form formulas for partial Bell polynomials at specific arguments of the sequences

$$1,0,1,0,9,0,225,0,\dots ,{[(2r-5)!!]}^2,0,{[(2r-3)!!]}^2,0,\dots$$

(13)

and

$$0,{\displaystyle \frac{1}{3}},0,{\displaystyle \frac{9}{5}},0,{\displaystyle \frac{225}{7}},0,1225,\dots ,0,{\displaystyle \frac{{[(2r-3)!!]}^2}{2r-1}},0,{\displaystyle \frac{{[(2r-1)!!]}^2}{2r+1}},\dots$$

(14)

for $r\in \mathbb{N}$, where

$$(-2m-1)!!={\displaystyle \frac{{(-1)}^m}{(2m-1)!!}}={(-1)}^m{\displaystyle \frac{2^{m}m!}{\left(2m\right)!}},m\in {\mathbb{N}}_0.$$

Theorem 4. 

For $r,k\in \mathbb{N}$, we have

$$\begin{array}{c}{B}_{2r+k,k}\left(1,0,1,0,9,0,225,0,\dots ,{[(2r-3)!!]}^2,0,{[(2r-1)!!]}^2\right)\hfill \\ =B_{2r+k,k}\left({\left(\arcsin t\right)}^{\prime }{{|}_{t=0},{\left(\arcsin t\right)}^{″}|}_{t=0},\dots ,{\left(\arcsin t\right)}^{(2r+1)}{|}_{t=0}\right)\hfill \\ ={(-1)}^k{B}_{2r+k,k}\left({\left(\arccos t\right)}^{\prime }{{|}_{t=0},{\left(\arccos t\right)}^{″}|}_{t=0},\dots ,{\left(\arccos t\right)}^{(2r+1)}{|}_{t=0}\right)\hfill \\ ={(-1)}^r{2}^{2r}Q(k,2r)\hfill \end{array}$$

(15)

and

$$\begin{array}{c}{B}_{2r+k-1,k}\left(1,0,1,0,9,0,225,0,\dots ,{[(2r-3)!!]}^2,0\right)\hfill \\ =B_{2r+k-1,k}\left({\left(\arcsin t\right)}^{\prime }{{|}_{t=0},{\left(\arcsin t\right)}^{″}|}_{t=0},\dots ,{\left(\arcsin t\right)}^{\left(2r\right)}{|}_{t=0}\right)\hfill \\ ={(-1)}^k{B}_{2r+k-1,k}\left({\left(\arccos t\right)}^{\prime }{{|}_{t=0},{\left(\arccos t\right)}^{″}|}_{t=0},\dots ,{\left(\arccos t\right)}^{\left(2r\right)}{|}_{t=0}\right)\hfill \\ =0,\hfill \end{array}$$

(16)

where $Q(k,2r)$ is given by (8).

Proof. 

It is well known that

$$\arcsin t=\sum _{n=0}^{\infty }{[(2n-1)!!]}^2{\displaystyle \frac{t^{2n+1}}{(2n+1)!}},\left|t\right|<1.$$

(17)

This means that

$${\left(\arcsin t\right)}^{(2n+1)}{|}_{t=0}={[(2n-1)!!]}^2=-{\left(\arccos t\right)}^{(2n+1)}{|}_{t=0}$$

(18)

for $n\in {\mathbb{N}}_0$ and

$${\left(\arcsin t\right)}^{\left(2n\right)}{|}_{t=0}=0=-{\left(\arccos t\right)}^{\left(2n\right)}{|}_{t=0}$$

(19)

for $n\in \mathbb{N}$.

In [29] [p. 60, 1.641], we find the series expansion

$$\arccos t={\displaystyle \frac{\pi }{2}}-\sum _{n=0}^{\infty }{\displaystyle \frac{(2n-1)!!}{\left(2n\right)!!}}{\displaystyle \frac{t^{2n+1}}{2n+1}},\left|t\right|<1.$$

(20)

The series expansion (20) and the equalities in (18) and (19) mean that

$${\left(\arccos t\right)}^{\left(2n\right)}{|}_{t=0}=0\mathrm{and}{\left(\arccos t\right)}^{(2n-1)}{|}_{t=0}=-{[(2n-3)!!]}^2$$

(21)

for $n\in \mathbb{N}$.

On [3] [p. 133], we see an identity

$${\displaystyle \frac{1}{m!}}{\left(\sum _{\ell =1}^{\infty }{x}_{\ell }{\displaystyle \frac{t^{\ell }}{\ell !}}\right)}^m=\sum _{n=m}^{\infty }{B}_{n,m}(x_1,x_2,\dots ,x_{n-m+1}){\displaystyle \frac{t^n}{n!}}$$

(22)

for $m\in {\mathbb{N}}_0$. Making use of the Formula (22) yields

$$B_{n+k,k}(x_1,x_2,\dots ,x_{n+1})=\left({\displaystyle \genfrac{}{}{0pt}{}{n+k}{k}}\right)\underset{t\to 0}{\lim }{\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}{\left[\sum _{\ell =0}^{\infty }{\displaystyle \frac{x_{\ell +1}}{(\ell +1)!}}{t}^{\ell }\right]}^k$$

(23)

for $n\ge k\in {\mathbb{N}}_0$. Taking $x_m={\left(\arccos t\right)}^{\left(m\right)}{|}_{t=0}$ for $m\in \mathbb{N}$ in (23), employing the values in (21), and utilizing the series expansion (7) in Theorem 1 give

$$\begin{array}{c}{B}_{n+k,k}\left({\left(\arccos t\right)}^{\prime }{|}_{t=0},{\left(\arccos t\right)}^{″}{|}_{t=0},\dots ,{\left(\arccos t\right)}^{(n+1)}{|}_{t=0}\right)\hfill \\ =B_{n+k,k}\left(-1,0,-1,0,-9,0,-225,\dots ,-{\displaystyle \frac{1-{(-1)}^{n+1}}{2}}{[(n-1)!!]}^2\right)\hfill \\ =\left({\displaystyle \genfrac{}{}{0pt}{}{n+k}{k}}\right)\underset{t\to 0}{\lim }{\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}{\left[\sum _{\ell =0}^{\infty }{\displaystyle \frac{{\left(\arccos t\right)}^{(\ell +1)}{|}_{t=0}}{(\ell +1)!}}{t}^{\ell }\right]}^k\hfill \\ ={(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n+k}{k}}\right)\underset{t\to 0}{\lim }{\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}{\left[{\displaystyle \frac{1}{t}}\sum _{\ell =1}^{\infty }{\displaystyle \frac{{\left(\arcsin t\right)}^{(\ell )}{|}_{t=0}}{\ell !}}{t}^{\ell }\right]}^k\hfill \\ ={(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n+k}{k}}\right)\underset{t\to 0}{\lim }{\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^k\hfill \\ ={(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n+k}{k}}\right)\underset{t\to 0}{\lim }{\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}\sum _{q=1}^{\infty }{\displaystyle \frac{{(-1)}^q}{\left(\genfrac{}{}{0pt}{}{k+2q}{k}\right)}}Q(k,2q){\displaystyle \frac{{\left(2t\right)}^{2q}}{\left(2q\right)!}}\hfill \\ ={(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n+k}{k}}\right)\underset{t\to 0}{\lim }\sum _{q=1}^{\infty }{\displaystyle \frac{{(-4)}^q}{\left(\genfrac{}{}{0pt}{}{k+2q}{k}\right)}}Q(k,2q){\langle 2q\rangle }_n{\displaystyle \frac{t^{2q-n}}{\left(2q\right)!}}\hfill \\ =\{\begin{array}{ll}{(-1)}^{k+r}{2}^{2r}Q(k,2r),& n=2r\\ 0,& n=2r-1\end{array}\hfill \end{array}$$

for $r\in \mathbb{N}$. Further employing the identity

$$B_{n,k}\left(\alpha \beta x_1,\alpha {\beta }^2{x}_2,\dots ,\alpha {\beta }^{n-k+1}{x}_{n-k+1}\right)={\alpha }^k{\beta }^n{B}_{n,k}(x_1,x_2,\dots ,x_{n-k+1})$$

(24)

for $n\ge k\in {\mathbb{N}}_0$ and $\alpha ,\beta \in \mathbb{C}$ in [2] [p. 412] and [3] [p. 135], and then simplifying them, we can determine the formulas in (15) and (16). □

Theorem 5. 

For $k,j\in {\mathbb{N}}_0$ such that $2j+1\ge k\in {\mathbb{N}}_0$, we have

$$\begin{array}{c}{B}_{2j+1,k}\left(0,{\displaystyle \frac{1}{3}},0,{\displaystyle \frac{9}{5}},0,{\displaystyle \frac{225}{7}},0,1225,\dots ,\dots ,{\displaystyle \frac{1+{(-1)}^k}{2}}{\displaystyle \frac{{[(2j-k+1)!!]}^2}{2j-k+3}}\right)\hfill \\ =B_{2j+1,k}\left({\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\prime }{|}_{t=0},{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{″}{|}_{t=0},\cdots ,{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{(2j-k+2)}{|}_{t=0}\right)\hfill \\ =0.\hfill \end{array}$$

(25)

For $k,j\in \mathbb{N}$ such that $2j\ge k\in \mathbb{N}$, we have

$$\begin{array}{c}{B}_{2j,k}\left(0,{\displaystyle \frac{1}{3}},0,{\displaystyle \frac{9}{5}},0,{\displaystyle \frac{225}{7}},0,1225,\dots ,\dots ,{\displaystyle \frac{1+{(-1)}^{k+1}}{2}}{\displaystyle \frac{{[(2j-k)!!]}^2}{2j-k+2}}\right)\hfill \\ =B_{2j,k}\left({\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\prime }{|}_{t=0},{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{″}{|}_{t=0},\cdots ,{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{(2j-k+1)}{|}_{t=0}\right)\hfill \\ ={(-1)}^{j+k}{\displaystyle \frac{\left(4j\right)!!}{(2j+k)!}}\sum _{q=1}^k{(-1)}^q\left({\displaystyle \genfrac{}{}{0pt}{}{2j+k}{k-q}}\right)Q(q,2j),\hfill \end{array}$$

(26)

where $Q(q,2j)$ is given by (8).

First Proof. 

The series expansion (17) can be rearranged as

$${\displaystyle \frac{\arcsin t}{t}}=\sum _{\ell =0}^{\infty }{\displaystyle \frac{{[(2\ell -1)!!]}^2}{2\ell +1}}{\displaystyle \frac{t^{2\ell }}{(2\ell )!}},\left|t\right|<1.$$

Therefore, for $q\in \mathbb{N}$, we have

$${\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\left(q\right)}{|}_{t=0}=\{\begin{array}{cc}0,\hfill & q=2p-1\hfill \\ {\displaystyle \frac{{[(2p-1)!!]}^2}{2p+1}},\hfill & q=2p\hfill \end{array}$$

(27)

for $p\in \mathbb{N}$. This is only the sequence listed in (14). Then, it is easy to see that

$$\begin{array}{c}{B}_{n,k}\left({\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\prime }{|}_{t=0},{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{″}{|}_{t=0},\dots ,{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{(n-k+1)}{|}_{t=0}\right)\hfill \\ =B_{n,k}\left(0,{\displaystyle \frac{1}{3}},0,{\displaystyle \frac{9}{5}},0,{\displaystyle \frac{225}{7}},0,1225,0,\dots ,{\displaystyle \frac{1+{(-1)}^{n-k+1}}{2}}{\displaystyle \frac{{[(n-k)!!]}^2}{n-k+2}}\right).\hfill \end{array}$$

Consequently, from [5] [Theorem 1.1] and [19] [Theorem 2], we can readily conclude Formulas (25) and (26). Theorem 5 is verified. □

Second Proof. 

Comparing (18) and (19) with (27) yields

$${\displaystyle \frac{{\left(\arcsin t\right)}^{\left(n\right)}{|}_{t=0}}{n}}={\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{(n-1)}{|}_{t=0},n\in \mathbb{N}.$$

(28)

Making use of (28) and employing the formula

$$B_{n,k}\left({\displaystyle \frac{x_2}{2}},{\displaystyle \frac{x_3}{3}},\dots ,{\displaystyle \frac{x_{n-k+2}}{n-k+2}}\right)={\displaystyle \frac{n!}{(n+k)!}}{B}_{n+k,k}(0,x_2,x_3,\dots ,x_{n+1})$$

in [3] [p. 136], we acquire

$$\begin{array}{c}{B}_{n,k}\left({\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\prime }{|}_{t=0},{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{″}{|}_{t=0},\dots ,{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{(n-k+1)}{|}_{t=0}\right)\hfill \\ =B_{n,k}\left({\displaystyle \frac{{\left(\arcsin t\right)}^{″}{|}_{t=0}}{2}},{\displaystyle \frac{{\left(\arcsin t\right)}^{‴}{|}_{t=0}}{3}},\dots ,{\displaystyle \frac{{\left(\arcsin t\right)}^{(n-k+2)}{|}_{t=0}}{n-k+2}}\right)\hfill \\ ={\displaystyle \frac{n!}{(n+k)!}}{B}_{n+k,k}\left(0,{\left(\arcsin t\right)}^{″}{|}_{t=0},{\left(\arcsin t\right)}^{‴}{|}_{t=0},\dots ,{\left(\arcsin t\right)}^{(n-k+1)}{|}_{t=0}\right).\hfill \end{array}$$

Repeating the proof of [19] [Theorem 2] results in the formulas in (26).

The formulas in (25) follow from the first formula in [5] [Theorem 1.1]. Theorem 5 is proved again. □

Third Proof. 

For $k\in \mathbb{N}$, utilizing the series expansion (7) in Theorem 1 leads to

$$\begin{array}{cc}{\left[{\displaystyle \frac{1}{t}}\left({\displaystyle \frac{\arcsin t}{t}}-1\right)\right]}^k\hfill & ={\displaystyle \frac{1}{t^k}}\sum _{j=0}^k{(-1)}^{k-j}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right){\left({\displaystyle \frac{\arcsin t}{t}}\right)}^j\hfill \\ & ={\displaystyle \frac{1}{t^k}}\left\{{(-1)}^k+\sum _{j=1}^k{(-1)}^{k-j}\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right)\left[1+\sum _{q=1}^{\infty }{\displaystyle \frac{{(-1)}^q}{\left(\genfrac{}{}{0pt}{}{j+2q}{j}\right)}}Q(j,2q){\displaystyle \frac{{\left(2t\right)}^{2q}}{\left(2q\right)!}}\right]\right\}\hfill \\ & ={\displaystyle \frac{{(-1)}^k}{t^k}}\sum _{q=1}^{\infty }{(-1)}^q\left[\sum _{j=1}^k{(-1)}^j{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{k}{j}\right)}{\left(\genfrac{}{}{0pt}{}{j+2q}{j}\right)}}Q(j,2q)\right]{\displaystyle \frac{{\left(2t\right)}^{2q}}{\left(2q\right)!}}.\hfill \end{array}$$

Hence, when $2\le 2q<k$, we derive a combinatorial identity:

$$\sum _{j=1}^k{(-1)}^j{\displaystyle \frac{Q(j,2q)}{(k-j)!(j+2q)!}}=0.$$

(29)

Accordingly, for $k,n\in \mathbb{N}$, we obtain

$$\begin{array}{cc}{\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}{\left[{\displaystyle \frac{1}{t}}\left({\displaystyle \frac{\arcsin t}{t}}-1\right)\right]}^k\hfill & ={(-1)}^k\sum _{q\ge k/2}^{\infty }{(-1)}^q\left[\sum _{j=1}^k{(-1)}^j{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{k}{j}\right)}{\left(\genfrac{}{}{0pt}{}{j+2q}{j}\right)}}Q(j,2q)\right]{\displaystyle \frac{2^{2q}}{\left(2q\right)!}}{\langle 2q-k\rangle }_n{t}^{2q-k-n}\hfill \\ & \to \{\begin{array}{cc}0,\hfill & \mathrm{if}1\le n<k\hfill \\ 0,\hfill & \mathrm{if}n+k\mathrm{is}\mathrm{odd}\hfill \\ {(-1)}^{(n+3k)/2}{\displaystyle \frac{2^{n+k}n!}{(n+k)!}}{\displaystyle \sum _{j=1}^k}{(-1)}^j{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{k}{j}\right)}{\left(\genfrac{}{}{0pt}{}{j+n+k}{j}\right)}}Q(j,n+k),\hfill & \mathrm{if}n+k\mathrm{is}\mathrm{even}\hfill \end{array}\hfill \end{array}$$

as $t\to 0$. Then, we conclude the following:

• If $k>n\in \mathbb{N}$, we acquire a combinatorial identity:

  $$\sum _{j=1}^k{(-1)}^j{\displaystyle \frac{Q(j,n+k)}{(k-j)!(j+n+k)!}}=0;$$

  (30)
• If $n=2r+k-1$ for $k,r\in \mathbb{N}$, we have

  $$\underset{t\to 0}{\lim }{\displaystyle \frac{{\mathrm{d}}^{2r+k-1}}{\mathrm{d}{t}^{2r+k-1}}}{\left[{\displaystyle \frac{1}{t}}\left({\displaystyle \frac{\arcsin t}{t}}-1\right)\right]}^k=0;$$

  (31)
• If $n=2r+k-2$ for $k,r\in \mathbb{N}$, we have

  $$\begin{array}{c}\underset{t\to 0}{\lim }{\displaystyle \frac{{\mathrm{d}}^{2r+k-2}}{\mathrm{d}{t}^{2r+k-2}}}{\left[{\displaystyle \frac{1}{t}}\left({\displaystyle \frac{\arcsin t}{t}}-1\right)\right]}^k\hfill \\ ={(-1)}^{r-1}{\displaystyle \frac{2^{2r+2k-2}(2r+k-2)!}{(2r+2k-2)!}}\sum _{j=1}^k{(-1)}^j{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{k}{j}\right)}{\left(\genfrac{}{}{0pt}{}{j+2r+2k-2}{j}\right)}}Q(j,2r+2k-2).\hfill \end{array}$$

  (32)

Setting

$$x_{\ell }={\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{(\ell )}{|}_{t=0},\ell \in \mathbb{N}$$

in (23) gives

$$\begin{array}{c}{B}_{n+k,k}\left({\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\prime }{|}_{t=0},{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{″}{|}_{t=0},\dots ,{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{(n+1)}{|}_{t=0}\right)\hfill \\ =\left({\displaystyle \genfrac{}{}{0pt}{}{n+k}{k}}\right)\underset{t\to 0}{\lim }{\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}{\left[\sum _{m=0}^{\infty }{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{(m+1)}{|}_{t=0}{\displaystyle \frac{t^m}{(m+1)!}}\right]}^k\hfill \\ ={(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n+k}{k}}\right)\underset{t\to 0}{\lim }{\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}{\left[{\displaystyle \frac{1}{t}}\sum _{m=1}^{\infty }{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\left(m\right)}{|}_{t=0}{\displaystyle \frac{t^m}{m!}}\right]}^k\hfill \\ ={(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n+k}{k}}\right)\underset{t\to 0}{\lim }{\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}{\left[{\displaystyle \frac{1}{t}}\left({\displaystyle \frac{\arcsin t}{t}}-1\right)\right]}^k\hfill \end{array}$$

for $n\ge k\in {\mathbb{N}}_0$. Replacing n by $2r+k-1$ for $k,r\in \mathbb{N}$ and utilizing the limit (31) reveal

$$\begin{array}{c}{B}_{2r+2k-1,k}\left({\left({\displaystyle \frac{\arcsin u}{u}}\right)}^{\prime }{|}_{u=0},{\left({\displaystyle \frac{\arcsin u}{u}}\right)}^{″}{|}_{u=0},\dots ,{\left({\displaystyle \frac{\arcsin u}{u}}\right)}^{(2r+k)}{|}_{u=0}\right)\hfill \\ ={(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{2r+2k-1}{k}}\right)\underset{u\to 0}{\lim }{\displaystyle \frac{{\mathrm{d}}^{2r+k-1}}{\mathrm{d}{u}^{2r+k-1}}}{\left[{\displaystyle \frac{1}{u}}\left({\displaystyle \frac{\arcsin u}{u}}-1\right)\right]}^k\hfill \\ =0.\hfill \end{array}$$

Substituting $2r+k-2$ with $k,r\in \mathbb{N}$ for n and employing the limit (32) result in

$$\begin{array}{c}{B}_{2r+2k-2,k}\left({\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\prime }{|}_{t=0},{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{″}{|}_{t=0},\dots ,{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{(2r+k-1)}{|}_{t=0}\right)\hfill \\ ={(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{2r+2k-2}{k}}\right)\underset{t\to 0}{\lim }{\displaystyle \frac{{\mathrm{d}}^{2r+k-2}}{\mathrm{d}{t}^{2r+k-2}}}{\left[{\displaystyle \frac{1}{t}}\left({\displaystyle \frac{\arcsin t}{t}}-1\right)\right]}^k\hfill \\ ={(-1)}^{k+r-1}{\displaystyle \frac{2^{2r+2k-2}}{k!}}\sum _{j=1}^k{(-1)}^j{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{k}{j}\right)}{\left(\genfrac{}{}{0pt}{}{j+2r+2k-2}{j}\right)}}Q(j,2r+2k-2).\hfill \end{array}$$

Theorem 5 is proved once again. □

3. Power Series Expansions for ${\mathbf{\left(}\mathbf{arccos}\mathit{t}\mathbf{\right)}}^{\mathit{\alpha }}$ and ${\mathbf{\left(}\mathbf{arccosh}\mathit{t}\mathbf{\right)}}^{\mathbf{2}\mathit{m}}$

By means of Formula (1), with the aid of explicit Formulas (15) and (16) in Theorem 4, and by virtue of combinatorial identities in [24] [Lemmas 2.1 and 2.2], we now establish power series expansions at the point $t=0$ for ${\left(\arccos t\right)}^{\alpha }$ and ${\left(\mathrm{arccosh}t\right)}^{2m}$ with $\alpha \in \mathbb{R}$ and $m\in \mathbb{N}$.

Theorem 6. 

For $\alpha \in \mathbb{R}$ and $\left|t\right|<1$, we have

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{2\arccos t}{\pi }}\right)}^{\alpha }=& 1+{\displaystyle \frac{{\langle \alpha \rangle }_2}{{\pi }^2}}{\displaystyle \frac{{\left(2t\right)}^2}{2!}}+\sum _{r=2}^{\infty }{(-1)}^r\left[\sum _{\ell =1}^r{(-1)}^{\ell }{\displaystyle \frac{{\langle \alpha \rangle }_{2\ell }}{{\pi }^{2\ell }}}Q(2\ell ,2r-2\ell )\right]{\displaystyle \frac{{\left(2t\right)}^{2r}}{\left(2r\right)!}}\hfill \\ & +\sum _{r=1}^{\infty }{(-1)}^r\left[\sum _{\ell =1}^r{(-1)}^{\ell -1}{\displaystyle \frac{{\langle \alpha \rangle }_{2\ell -1}}{{\pi }^{2\ell -1}}}Q(2\ell -1,2r-2\ell )\right]{\displaystyle \frac{{\left(2t\right)}^{2r-1}}{(2r-1)!}},\hfill \end{array}$$

(33)

where ${\langle \alpha \rangle }_r$ for $\alpha \in \mathbb{R}$ and $r\in \mathbb{N}$ stands for the falling factorials defined by (3), and the quantity $Q(m,n)$ is defined by (8).

Proof. 

Let $u=u\left(t\right)=\arccos t$. It is clear that $u=u\left(t\right)=\arccos t\to {\displaystyle \frac{\pi }{2}}$ as $t\to 0$. By means of the Faà di Bruno Formula (1) and the values in (21), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\mathrm{d}}^m\left[{\left(\arccos t\right)}^{\alpha }\right]}{\mathrm{d}{t}^m}}& =\sum _{q=1}^m{\displaystyle \frac{{\mathrm{d}}^q\left(u^{\alpha }\right)}{\mathrm{d}{u}^q}}{B}_{m,q}\left({\left(\arccos t\right)}^{\prime },{\left(\arccos t\right)}^{″},\dots ,{\left(\arccos t\right)}^{(m-q+1)}\right)\hfill \\ & =\sum _{q=1}^m{\langle \alpha \rangle }_q{u}^{\alpha -q}{B}_{m,q}\left({\left(\arccos t\right)}^{\prime },{\left(\arccos t\right)}^{″},\dots ,{\left(\arccos t\right)}^{(m-q+1)}\right)\hfill \\ & \to \sum _{q=1}^m{\langle \alpha \rangle }_q{\left({\displaystyle \frac{\pi }{2}}\right)}^{\alpha -q}{B}_{m,q}\left(-1,0,-1,0,-9,\dots ,-{\displaystyle \frac{1-{(-1)}^{m-q+1}}{2}}{[(m-q-1)!!]}^2\right)\hfill \end{array}$$

as $t\to 0$ for $m\in \mathbb{N}$.

When $m=2r$ and $r\ge 2$, it follows that

$$\begin{array}{cc}\hfill \underset{t\to 0}{\lim }{\displaystyle \frac{{\mathrm{d}}^{2r}\left[{\left(\arccos t\right)}^{\alpha }\right]}{\mathrm{d}{t}^{2r}}}& =\sum _{\ell =1}^{2r}{\langle \alpha \rangle }_{\ell }{\left({\displaystyle \frac{\pi }{2}}\right)}^{\alpha -\ell }{B}_{2r,\ell }\left(-1,0,-1,0,-9,\dots ,-{\displaystyle \frac{1-{(-1)}^{1-\ell }}{2}}{[(2r-\ell -1)!!]}^2\right)\hfill \\ & =\sum _{\ell =1}^r{\langle \alpha \rangle }_{2\ell }{\left({\displaystyle \frac{\pi }{2}}\right)}^{\alpha -2\ell }{B}_{2r,2\ell }\left(1,0,1,0,9,0,225,\dots ,0,{[(2r-2\ell -1)!!]}^2\right)\hfill \\ & =\sum _{\ell =1}^r{(-1)}^{r-\ell }{\langle \alpha \rangle }_{2\ell }{\pi }^{\alpha -2\ell }{2}^{2r-\alpha }Q(2\ell ,2r-2\ell ),\hfill \end{array}$$

(34)

where we used the identity (24) and the formulas in (15) and (16) in Theorem 4.

It is easy to see that

$$\underset{t\to 0}{\lim }{\displaystyle \frac{{\mathrm{d}}^2\left[{\left(\arccos t\right)}^{\alpha }\right]}{\mathrm{d}{t}^2}}=(\alpha -1)\alpha {\left({\displaystyle \frac{\pi }{2}}\right)}^{\alpha -2}.$$

(35)

When $m=2r-1$ for $r\ge 2$, it follows that

$$\begin{array}{cc}\hfill \underset{t\to 0}{\lim }{\displaystyle \frac{{\mathrm{d}}^{2r-1}\left[{\left(\arccos t\right)}^{\alpha }\right]}{\mathrm{d}{t}^{2r-1}}}& =\sum _{\ell =1}^{2r-1}{(-1)}^{\ell }{\langle \alpha \rangle }_{\ell }{\left({\displaystyle \frac{\pi }{2}}\right)}^{\alpha -\ell }{B}_{2r-1,\ell }\left(1,0,1,0,9,\dots ,{\displaystyle \frac{1-{(-1)}^{\ell }}{2}}{[(2r-\ell -2)!!]}^2\right)\hfill \\ & =-\sum _{\ell =1}^r{\langle \alpha \rangle }_{2\ell -1}{\left({\displaystyle \frac{\pi }{2}}\right)}^{\alpha -2\ell +1}{B}_{2r-1,2\ell -1}\left(1,0,1,0,9,\dots ,{[(2r-2\ell -1)!!]}^2\right)\hfill \\ & =\sum _{\ell =1}^r{(-1)}^{r-\ell -1}{\langle \alpha \rangle }_{2\ell -1}{\pi }^{\alpha -2\ell +1}{2}^{2r-\alpha -1}Q(2\ell -1,2r-2\ell ),\hfill \end{array}$$

(36)

where we used the identity (24) and those formulas in (15) and (16) in Theorem 4.

It is not difficult to find that

$$\underset{t\to 0}{\lim }{\displaystyle \frac{\mathrm{d}\left[{\left(\arccos t\right)}^{\alpha }\right]}{\mathrm{d}t}}=-\alpha {\left({\displaystyle \frac{\pi }{2}}\right)}^{\alpha -1}.$$

(37)

Combining the four limits (34), (35), (36), and (37) and then simplifying them yield the series expansion (33). □

Corollary 1. 

For $\left|t\right|<1$, we have

$${\left({\displaystyle \frac{2\arccos t}{\pi }}\right)}^2=1-{\displaystyle \frac{4}{\pi }}\sum _{r=1}^{\infty }{[(2r-3)!!]}^2{\displaystyle \frac{t^{2r-1}}{(2r-1)!}}+{\displaystyle \frac{8}{{\pi }^2}}\sum _{r=1}^{\infty }{[(2r-2)!!]}^2{\displaystyle \frac{t^{2r}}{\left(2r\right)!}}.$$

(38)

Proof. 

Setting $\alpha =2$ in the series expansion (33) in Theorem 6 arrives at

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{2\arccos t}{\pi }}\right)}^2=& 1+{\displaystyle \frac{2}{{\pi }^2}}{\displaystyle \frac{{\left(2t\right)}^2}{2!}}+{\displaystyle \frac{2}{{\pi }^2}}\sum _{r=2}^{\infty }{(-1)}^{r+1}Q(2,2r-2){\displaystyle \frac{{\left(2t\right)}^{2r}}{\left(2r\right)!}}\hfill \\ & +{\displaystyle \frac{2}{\pi }}\sum _{r=1}^{\infty }{(-1)}^{r}Q(1,2r-2){\displaystyle \frac{{\left(2t\right)}^{2r-1}}{(2r-1)!}}.\hfill \end{array}$$

Further, by using the identities

$$Q(2,2m)=\sum _{n=0}^{2m}(n+1)s(2m+1,n+1)m^n={(-1)}^m{(m!)}^2,m\in \mathbb{N}$$

(39)

and

$$Q(1,2m)=\sum _{\ell =0}^{2k}s(2k,\ell ){\left(k-{\displaystyle \frac{1}{2}}\right)}^{\ell }={(-1)}^k{\left[{\displaystyle \frac{(2k-1)!!}{2^k}}\right]}^2,k\in \mathbb{N}$$

(40)

in [24] [Lemmas 2.1 and 2.2], we acquire

$${\left({\displaystyle \frac{2\arccos t}{\pi }}\right)}^2=1+{\displaystyle \frac{2}{{\pi }^2}}{\displaystyle \frac{{\left(2t\right)}^2}{2!}}+{\displaystyle \frac{2}{{\pi }^2}}\sum _{r=2}^{\infty }{[(r-1)!]}^2{\displaystyle \frac{{\left(2t\right)}^{2r}}{\left(2r\right)!}}-{\displaystyle \frac{2}{\pi }}\sum _{r=1}^{\infty }{\left[{\displaystyle \frac{(2r-3)!!}{2^{r-1}}}\right]}^2{\displaystyle \frac{{\left(2t\right)}^{2r-1}}{(2r-1)!}}.$$

The series expansion (38) thus follows. We proved Corollary 1. □

Corollary 2. 

For $k\in \mathbb{N}$ and $\left|t\right|<1$, we have

$${\left({\displaystyle \frac{2\mathrm{arccosh}t}{\pi }}\right)}^2=-1+{\displaystyle \frac{4}{\pi }}\sum _{r=1}^{\infty }{[(2r-3)!!]}^2{\displaystyle \frac{t^{2r-1}}{(2r-1)!}}-{\displaystyle \frac{8}{{\pi }^2}}\sum _{r=1}^{\infty }{[(2r-2)!!]}^2{\displaystyle \frac{t^{2r}}{\left(2r\right)!}}$$

(41)

and

$$\begin{array}{cc}\hfill {(-1)}^k{\left({\displaystyle \frac{2\mathrm{arccosh}t}{\pi }}\right)}^{2k}=& 1+{\displaystyle \frac{{\langle 2k\rangle }_2}{{\pi }^2}}{\displaystyle \frac{{\left(2t\right)}^2}{2!}}+\sum _{r=2}^{\infty }{(-1)}^r\left[\sum _{\ell =1}^r{(-1)}^{\ell }{\displaystyle \frac{{\langle 2k\rangle }_{2\ell }}{{\pi }^{2\ell }}}Q(2\ell ,2r-2\ell )\right]{\displaystyle \frac{{\left(2t\right)}^{2r}}{\left(2r\right)!}}\hfill \\ & +\sum _{r=1}^{\infty }{(-1)}^r\left[\sum _{\ell =1}^r{(-1)}^{\ell -1}{\displaystyle \frac{{\langle 2k\rangle }_{2\ell -1}}{{\pi }^{2\ell -1}}}Q(2\ell -1,2r-2\ell )\right]{\displaystyle \frac{{\left(2t\right)}^{2r-1}}{(2r-1)!}},\hfill \end{array}$$

(42)

where the falling factorials ${\langle 2k\rangle }_r$ for $k,r\in \mathbb{N}$ are defined by (3) and the quantity $Q(m,n)$ is defined by (8).

Proof. 

This follows from substituting the relation $\arccos t=-\mathrm{i}\mathrm{arccosh}t$ into the series expansions (38) and (6) for $\alpha =2k$. □

4. Power Series Expansions for ${\left({\displaystyle \frac{\mathbf{arcsin}\mathit{t}}{\mathit{t}}}\right)}^{\mathit{\alpha }}$ and ${\left({\displaystyle \frac{\mathbf{arcsinh}\mathit{t}}{\mathit{t}}}\right)}^{\mathbf{2}\mathit{m}}$

In this section, in light of Formula (1) and with the help of explicit formulas in (25) and (26) in Theorem 5, we establish power series expansions around the point $t=0$ for ${\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\alpha }$ and ${\left({\displaystyle \frac{\mathrm{arcsinh}t}{t}}\right)}^{2m}$ with $\alpha \in \mathbb{R}$ and $m\in \mathbb{N}$.

Theorem 7. 

For $\alpha \in \mathbb{R}$ and $\left|t\right|<1$, we have

$${\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\alpha }=1+\sum _{n=1}^{\infty }{(-1)}^n\left[\sum _{k=1}^{2n}{(-1)}^k{\displaystyle \frac{{\langle \alpha \rangle }_k}{(2n+k)!}}\sum _{q=1}^k{(-1)}^q\left({\displaystyle \genfrac{}{}{0pt}{}{2n+k}{k-q}}\right)Q(q,2n)\right]{\left(2t\right)}^{2n},$$

(43)

where ${\langle \alpha \rangle }_k$ is defined by (3) and $Q(q,2n)$ is given by (8).

Proof. 

By the Faà di Bruno Formula (1), it follows that

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}\left[{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\alpha }\right]& =\sum _{k=0}^n{\langle \alpha \rangle }_k{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\alpha -k}{B}_{n,k}\left({\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\prime },{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{″},\dots ,{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{(n-k+1)}\right)\hfill \\ & \to \sum _{k=1}^n{\langle \alpha \rangle }_k{B}_{n,k}\left({\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\prime }{|}_{t=0},{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{″}{|}_{t=0},\dots ,{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{(n-k+1)}{|}_{t=0}\right)\hfill \end{array}$$

as $x\to 0$ for $n\in \mathbb{N}$. Consequently, with the help of explicit formulas in (25) and (26) in Theorem 5, we can determine that

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\alpha }& =\sum _{n=0}^{\infty }\underset{t\to 0}{\lim }{\displaystyle \frac{{\mathrm{d}}^{2n}}{\mathrm{d}{t}^{2n}}}\left[{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\alpha }\right]{\displaystyle \frac{t^{2n}}{\left(2n\right)!}}\hfill \\ & =1+\sum _{n=1}^{\infty }\left[\sum _{k=1}^{2n}{\langle \alpha \rangle }_k{B}_{2n,k}\left({\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{\prime }{|}_{t=0},{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{″}{|}_{t=0},\dots ,{\left({\displaystyle \frac{\arcsin t}{t}}\right)}^{(2n-k+1)}{|}_{t=0}\right)\right]{\displaystyle \frac{t^{2n}}{\left(2n\right)!}}\hfill \\ & =1+\sum _{n=1}^{\infty }{(-1)}^n\left(4n\right)!!\left[\sum _{k=1}^{2n}{(-1)}^k{\displaystyle \frac{{\langle \alpha \rangle }_k}{(2n+k)!}}\sum _{q=1}^k{(-1)}^q\left({\displaystyle \genfrac{}{}{0pt}{}{2n+k}{k-q}}\right)Q(q,2n)\right]{\displaystyle \frac{t^{2n}}{\left(2n\right)!}}.\hfill \end{array}$$

Theorem 7 is thus proved. □

Theorem 8. 

For $m\in \mathbb{N}$ and $\left|t\right|<1$, we have

$${\left({\displaystyle \frac{\mathrm{arcsinh}t}{t}}\right)}^{2m}=1+\sum _{n=1}^{\infty }\left[\sum _{k=1}^{2n}{(-1)}^k{\displaystyle \frac{{\langle 2m\rangle }_k}{(2n+k)!}}\sum _{q=1}^k{(-1)}^q\left({\displaystyle \genfrac{}{}{0pt}{}{2n+k}{k-q}}\right)Q(q,2n)\right]{\left(2t\right)}^{2n},$$

(44)

where $Q(q,2n)$ is given by (8).

Proof. 

The series expansion (44) follows from applying the relation

$$\arcsin t=-\mathrm{i}\mathrm{arcsinh}\left(\mathrm{i}t\right)$$

to (43) in Theorem 7, replacing t by $\mathrm{i}t$, and rearranging. □

5. Infinite Series Representations for $\mathit{\pi }$ and Its Powers

There has been a wide variety of remarkable series, product, and integral representations for the circular constant $\pi$ and its relatives. For detailed information on this subject, we refer to [30,31,32] and the references cited therein.

By comparing the series expansion (11) for $k=1$ in Theorem 3 with the series expansion (38) in Corollary 1, we can find the following infinite series representations of $\pi$ and ${\pi }^2$.

Theorem 9. 

For $r\in \mathbb{N}$, the constants π and ${\pi }^2$ can be represented by

$$\pi ={\displaystyle \frac{2^{2r-1}}{\left(\genfrac{}{}{0pt}{}{2r-2}{r-1}\right)}}\sum _{m=2r-1}^{\infty }{\displaystyle \frac{2^m}{m}}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{m-1}{2r-2}\right)}{\left(\genfrac{}{}{0pt}{}{2m}{m}\right)}}$$

(45)

and

$${\displaystyle \frac{{\pi }^2}{8}}=\sum _{m=1}^{\infty }{\displaystyle \frac{2^m}{m^2}}{\displaystyle \frac{1}{\left(\genfrac{}{}{0pt}{}{2m}{m}\right)}}.$$

(46)

Proof. 

For $\left|t\right|<1$, the power series expansion (38) can be reformulated as

$${\left(\arccos t\right)}^2={\displaystyle \frac{{\pi }^2}{4}}-\pi \sum _{r=1}^{\infty }{[(2r-3)!!]}^2{\displaystyle \frac{t^{2r-1}}{(2r-1)!}}+2\sum _{r=1}^{\infty }{[(2r-2)!!]}^2{\displaystyle \frac{t^{2r}}{\left(2r\right)!}}.$$

(47)

The series expansion (11) for $k=1$ in Theorem 3 can be rearranged as

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\left(\arccos t\right)}^2}{2!}}& =1-t+\sum _{m=1}^{\infty }{\displaystyle \frac{m!}{(2m+1)!!(m+1)}}\sum _{\ell =0}^{m+1}{(-1)}^{\ell }\left({\displaystyle \genfrac{}{}{0pt}{}{m+1}{\ell }}\right)t^{\ell }\hfill \\ & =\sum _{m=0}^{\infty }{\displaystyle \frac{m!}{(2m+1)!!(m+1)}}-t\sum _{m=0}^{\infty }{\displaystyle \frac{m!}{(2m+1)!!}}+\sum _{m=1}^{\infty }{\displaystyle \frac{m!}{(2m+1)!!(m+1)}}\sum _{\ell =1}^m{(-1)}^{\ell +1}\left({\displaystyle \genfrac{}{}{0pt}{}{m+1}{\ell +1}}\right)t^{\ell +1}\hfill \\ & =\sum _{m=0}^{\infty }{\displaystyle \frac{m!}{(2m+1)!!(m+1)}}-t\sum _{m=0}^{\infty }{\displaystyle \frac{m!}{(2m+1)!!}}+\sum _{\ell =2}^{\infty }{(-1)}^{\ell }\left[\sum _{m=\ell }^{\infty }{\displaystyle \frac{(m-1)!}{(2m-1)!!m}}\left({\displaystyle \genfrac{}{}{0pt}{}{m}{\ell }}\right)\right]t^{\ell }\hfill \\ & =\sum _{m=0}^{\infty }{\displaystyle \frac{2^{m+1}}{{(m+1)}^2}}{\displaystyle \frac{1}{\left(\genfrac{}{}{0pt}{}{2m+2}{m+1}\right)}}+\sum _{\ell =1}^{\infty }{(-1)}^{\ell }\left[\sum _{m=\ell }^{\infty }{\displaystyle \frac{2^m}{m}}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{m-1}{\ell -1}\right)}{\left(\genfrac{}{}{0pt}{}{2m}{m}\right)}}\right]{\displaystyle \frac{t^{\ell }}{\ell }}.\hfill \end{array}$$

Accordingly, we obtain

$${\left(\arccos t\right)}^2=2\sum _{q=0}^{\infty }{\displaystyle \frac{2^{q+1}}{{(q+1)}^2}}{\displaystyle \frac{1}{\left(\genfrac{}{}{0pt}{}{2q+2}{q+1}\right)}}+2\sum _{\ell =1}^{\infty }{(-1)}^{\ell }\left[\sum _{q=\ell }^{\infty }{\displaystyle \frac{2^q}{q}}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{q-1}{\ell -1}\right)}{\left(\genfrac{}{}{0pt}{}{2q}{q}\right)}}\right]{\displaystyle \frac{t^{\ell }}{\ell }}.$$

(48)

Comparing the series expansion (48) with the series expansion (47) produces

$$\begin{array}{cc}\hfill 2\sum _{m=0}^{\infty }{\displaystyle \frac{2^{m+1}}{{(m+1)}^2}}{\displaystyle \frac{1}{\left(\genfrac{}{}{0pt}{}{2m+2}{m+1}\right)}}& ={\displaystyle \frac{{\pi }^2}{4}},\hfill \\ \hfill 2{(-1)}^{2r-1}\left[\sum _{m=2r-1}^{\infty }{\displaystyle \frac{2^m}{m}}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{m-1}{2r-2}\right)}{\left(\genfrac{}{}{0pt}{}{2m}{m}\right)}}\right]{\displaystyle \frac{t^{2r-1}}{2r-1}}& =-\pi {[(2r-3)!!]}^2{\displaystyle \frac{t^{2r-1}}{(2r-1)!}},\hfill \end{array}$$

and

$$2{(-1)}^{2r}\left[\sum _{m=2r}^{\infty }{\displaystyle \frac{2^m}{m}}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{m-1}{2r-1}\right)}{\left(\genfrac{}{}{0pt}{}{2m}{m}\right)}}\right]{\displaystyle \frac{t^{2r}}{2r}}=2{[(2r-2)!!]}^2{\displaystyle \frac{t^{2r}}{\left(2r\right)!}}$$

(49)

for $r\in \mathbb{N}$, in which the second equation can be rewritten as

$$\sum _{m=2r-1}^{\infty }{\displaystyle \frac{2^m}{m}}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{m-1}{2r-2}\right)}{\left(\genfrac{}{}{0pt}{}{2m}{m}\right)}}={\displaystyle \frac{\pi }{2}}{\displaystyle \frac{{[(2r-3)!!]}^2}{(2r-2)!}}=\pi {\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2r-2}{r-1}\right)}{2^{2r-1}}}$$

for $r\in \mathbb{N}$. The series representations (45) and (46) follow. □

By taking the special value $x={\displaystyle \frac{1}{2}}$ on both sides of (33) in Theorem 6, we can obtain the following interesting series representation.

Theorem 10. 

For $\alpha \in \mathbb{R}$, we have

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{2}{3}}\right)}^{\alpha }=& 1+\sum _{\ell =1}^{\infty }{(-1)}^{\ell }\left[\sum _{j=\ell }^{\infty }{(-1)}^j{\displaystyle \frac{Q(2\ell -1,2j-2\ell )}{\left(2j\right)!}}\right]{\displaystyle \frac{{\langle \alpha \rangle }_{2\ell }}{{\pi }^{2\ell }}}\hfill \\ & +\sum _{\ell =1}^{\infty }{(-1)}^{\ell -1}\left[\sum _{j=\ell }^{\infty }{(-1)}^j{\displaystyle \frac{Q(2\ell -1,2j-2\ell )}{(2j-1)!}}\right]{\displaystyle \frac{{\langle \alpha \rangle }_{2\ell -1}}{{\pi }^{2\ell -1}}},\hfill \end{array}$$

(50)

where $Q(m,n)$ is given by (8).

Proof. 

This follows from taking the special value $t={\displaystyle \frac{1}{2}}$ on both sides of (33) in Theorem 6 and interchanging the orders of sums. □

Corollary 3. 

The circular constant π can be represented as

$${\displaystyle \frac{\pi }{6}}=\sum _{r=1}^{\infty }{\displaystyle \frac{{[(2r-3)!!]}^2}{(4r-2)!!}}.$$

(51)

Proof. 

By taking $\alpha =1$ on both sides of (50) in Theorem 10 and then simplifying them, we discover

$${\displaystyle \frac{\pi }{3}}=\sum _{r=1}^{\infty }{\displaystyle \frac{{(-1)}^{r-1}}{(2r-1)!}}\sum _{q=0}^{2r-2}s(2r-2,q){\left({\displaystyle \frac{2r-3}{2}}\right)}^q.$$

Further, by making use of the identity (40) and simplifying it, we conclude the series representation (51). The proof of Corollary 3 is complete. □

Corollary 4 

([24] [Remark 3.1] and [33] [p. 453, (14)]). The constant ${\pi }^2$ satisfies

$${\displaystyle \frac{{\pi }^2}{18}}=\sum _{r=1}^{\infty }{\displaystyle \frac{{[(r-1)!]}^2}{\left(2r\right)!}}.$$

(52)

Proof. 

The series representation (53) comes from taking $\alpha =2$ on both sides of (50) in Theorem 10 and then simplifying, thus giving

$${\displaystyle \frac{4}{9}}=1-{\displaystyle \frac{2}{{\pi }^2}}\sum _{r=1}^{\infty }{\displaystyle \frac{{(-1)}^r}{\left(2r\right)!}}Q(2,2r-2)+{\displaystyle \frac{2}{\pi }}\sum _{r=1}^{\infty }{\displaystyle \frac{{(-1)}^r}{(2r-1)!}}Q(1,2r-2).$$

Further, using the formula $s(j,1)={(-1)}^{j-1}(j-1)!$ for $j\in \mathbb{N}$, employing the identities (39) and (40), and simplifying them, we acquire

$${\displaystyle \frac{4}{9}}=1-{\displaystyle \frac{2}{{\pi }^2}}\sum _{r=1}^{\infty }{\displaystyle \frac{{(-1)}^r}{\left(2r\right)!}}{(-1)}^{r-1}{[(r-1)!]}^2+{\displaystyle \frac{2}{\pi }}\sum _{r=1}^{\infty }{\displaystyle \frac{{(-1)}^r}{(2r-1)!}}{(-1)}^{r-1}{\left[{\displaystyle \frac{(2r-3)!!}{2^{r-1}}}\right]}^2,$$

that is,

$${\displaystyle \frac{4}{9}}=1-{\displaystyle \frac{4}{\pi }}\sum _{r=1}^{\infty }{\displaystyle \frac{{[(2r-3)!!]}^2}{(4r-2)!!}}+{\displaystyle \frac{2}{{\pi }^2}}\sum _{r=1}^{\infty }{\displaystyle \frac{{[(r-1)!]}^2}{\left(2r\right)!}}.$$

(53)

Substituting (51) in Corollary 3 into (53) reveals

$${\displaystyle \frac{4}{9}}=1-{\displaystyle \frac{4}{\pi }}{\displaystyle \frac{\pi }{6}}+{\displaystyle \frac{2}{{\pi }^2}}\sum _{r=1}^{\infty }{\displaystyle \frac{{[(r-1)!]}^2}{\left(2r\right)!}}.$$

The series representation (52) is thus obtained. The proof of Corollary 4 is complete. □

Theorem 11. 

For $\alpha \in \mathbb{R}$, we have

$${\left({\displaystyle \frac{\pi }{3}}\right)}^{\alpha }=1+\sum _{n=1}^{\infty }{(-1)}^n\left[\sum _{k=1}^{2n}{(-1)}^k{\displaystyle \frac{{\langle \alpha \rangle }_k}{(2n+k)!}}\sum _{q=1}^k{(-1)}^q\left({\displaystyle \genfrac{}{}{0pt}{}{2n+k}{k-q}}\right)Q(q,2n)\right],$$

(54)

where ${\langle \alpha \rangle }_k$ is defined by (3) and $Q(q,2n)$ is given by (8).

Proof. 

Letting $x={\displaystyle \frac{1}{2}}$ in (43) leads to (54) in Theorem 11. □

6. Several Combinatorial Identities

Besides combinatorial identities in (29) and (30), squaring on both sides of the Equation (20), comparing with the power series expansion (38) in Corollary 1, and then rewriting the Equation (49), we can derive the following several combinatorial identities.

Theorem 12. 

For $r\in \mathbb{N}$, we have

$$\sum _{j=0}^{r-1}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2j}{j}\right)}{2j+1}}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{2(r-j-1)}{r-j-1}\right)}{2r-2j-1}}={\displaystyle \frac{2^{4r-3}}{r^2\left(\genfrac{}{}{0pt}{}{2r}{r}\right)}}$$

(55)

and

$$\sum _{m=2r}^{\infty }{\displaystyle \frac{2^m}{m}}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{m-1}{2r-1}\right)}{\left(\genfrac{}{}{0pt}{}{2m}{m}\right)}}={\displaystyle \frac{2^{2r-2}}{2r-1}}{\displaystyle \frac{1}{\left(\genfrac{}{}{0pt}{}{2(r-1)}{r-1}\right)}}.$$

(56)

Proof. 

From the series expansion (20) and Cauchy’s product, we conclude that

$$\begin{array}{cc}\hfill {\left(\arccos t\right)}^2& ={\left[{\displaystyle \frac{\pi }{2}}-\sum _{r=0}^{\infty }{\displaystyle \frac{(2r-1)!!}{\left(2r\right)!!}}{\displaystyle \frac{t^{2r+1}}{2r+1}}\right]}^2\hfill \\ & ={\left({\displaystyle \frac{\pi }{2}}\right)}^2-\pi \sum _{r=0}^{\infty }{\displaystyle \frac{(2r-1)!!}{\left(2r\right)!!}}{\displaystyle \frac{t^{2r+1}}{2r+1}}+t^2{\left[\sum _{r=0}^{\infty }{\displaystyle \frac{(2r-1)!!}{\left(2r\right)!!}}{\displaystyle \frac{t^{2r}}{2r+1}}\right]}^2\hfill \\ & ={\left({\displaystyle \frac{\pi }{2}}\right)}^2-\pi \sum _{r=0}^{\infty }{\displaystyle \frac{(2r-1)!!}{\left(2r\right)!!}}{\displaystyle \frac{t^{2r+1}}{2r+1}}+\sum _{r=0}^{\infty }\left[\sum _{q=0}^r{\displaystyle \frac{(2q-1)!!}{\left(2q\right)!!(2q+1)}}{\displaystyle \frac{(2r-2q-1)!!}{(2r-2q)!!(2r-2q+1)}}\right]t^{2r+2}\hfill \\ & ={\left({\displaystyle \frac{\pi }{2}}\right)}^2-\pi \sum _{r=1}^{\infty }{\displaystyle \frac{(2r-3)!!}{(2r-2)!!}}{\displaystyle \frac{t^{2r-1}}{2r-1}}+\sum _{r=1}^{\infty }\left[\sum _{q=0}^{r-1}{\displaystyle \frac{(2q-1)!!}{\left(2q\right)!!(2q+1)}}{\displaystyle \frac{(2r-2q-3)!!}{(2r-2q-2)!!(2r-2q-1)}}\right]t^{2r}.\hfill \end{array}$$

Comparing this with (47), equating coefficients of the factors $t^{2r}$, and then simplifying them reveal the combinatorial identity (55).

Equation (49) can be simplified as

$$\sum _{n=2r}^{\infty }{\displaystyle \frac{2^n}{n}}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{n-1}{2r-1}\right)}{\left(\genfrac{}{}{0pt}{}{2n}{n}\right)}}={\displaystyle \frac{{[(2r-2)!!]}^2}{(2r-1)!}}={\displaystyle \frac{2^{2r-2}}{2r-1}}{\displaystyle \frac{1}{\left(\genfrac{}{}{0pt}{}{2r-2}{r-1}\right)}},r\in \mathbb{N}.$$

The combinatorial identity (56) is thus proved. □

Theorem 13. 

For $m,n\in \mathbb{N}$, we have

$$\sum _{k=1}^{2n}{(-1)}^k{\displaystyle \frac{{\langle m\rangle }_k}{(2n+k)!}}\sum _{q=1}^k{(-1)}^q\left({\displaystyle \genfrac{}{}{0pt}{}{2n+k}{k-q}}\right)Q(q,2n)={\displaystyle \frac{m!}{(m+2n)!}}Q(m,2n).$$

Proof. 

This follows from comparing the series expansion (7) in Theorem 1 with the series expansion (43) in Theorem 7 and equating coefficients of the terms ${\left(2t\right)}^n$ for $n\in \mathbb{N}$. □

7. Useful Remarks

Finally, we give several useful remarks related to our main results.

Remark 1. 

It is trivial that $B_{k,k}\left(t\right)=t^k$ and $B_{k+1,0}(x_1,x_2,\dots ,x_{k+2})=0$ for $k\in {\mathbb{N}}_0$. Consequently, we considered in Theorem 4 all nontrivial cases of partial Bell polynomials $B_{n,k}$ for $n\ge k\in {\mathbb{N}}_0$ with respect to the sequence listed in (13).

Remark 2. 

Using Formulas (15) and (16) in Theorem 4, we can compute the power series expansions around $t=0$ for the kind of composite functions $f\left(\arccos t\right)$ and $f\left(\arcsin t\right)$, if the derivatives of f are explicitly computable.

Remark 3. 

The first identity (25) in Theorem 5 is a special case of the following general conclusion obtained in [5] [Theorem 1.1]: For $k,n\in {\mathbb{N}}_0$ and $x_m\in \mathbb{C}$ with $m\in \mathbb{N}$, we have

$$B_{2n+1,k}\left(0,x_2,0,x_4,\dots ,{\displaystyle \frac{1+{(-1)}^k}{2}}{x}_{2n-k+2}\right)=0.$$

The second identity (26) in Theorem 5 is recovered [19] [Theorem 2].

Employing Formulas (25) and (26) in Theorem 5, we can discover power series expansions around $t=0$ for the kind of composite functions $f\left({\displaystyle \frac{\arcsin t}{t}}\right)$, if the derivatives of f is explicitly computable.

Remark 4. 

In the series expansion (33), if $0^0$ is assumed to be 1, the term ${\displaystyle \frac{{\langle \alpha \rangle }_2}{{\pi }^2}}{\displaystyle \frac{{\left(2t\right)}^2}{2!}}$ can be combined into the first sum in Theorem 6. Then, the series expansion (33) can be reformulated as

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{2\arccos t}{\pi }}\right)}^{\alpha }=& 1+\sum _{r=1}^{\infty }{(-1)}^r\left[\sum _{\ell =1}^r{(-1)}^{\ell }{\displaystyle \frac{{\langle \alpha \rangle }_{2\ell }}{{\pi }^{2\ell }}}Q(2\ell ,2r-2\ell )\right]{\displaystyle \frac{{\left(2t\right)}^{2r}}{\left(2r\right)!}}\hfill \\ & +\sum _{r=1}^{\infty }{(-1)}^r\left[\sum _{\ell =1}^r{(-1)}^{\ell -1}{\displaystyle \frac{{\langle \alpha \rangle }_{2\ell -1}}{{\pi }^{2\ell -1}}}Q(2\ell -1,2r-2\ell )\right]{\displaystyle \frac{{\left(2t\right)}^{2r-1}}{(2r-1)!}}\hfill \end{array}$$

for $\alpha \in \mathbb{R}$ and $\left|t\right|<1$. Similarly, the series expansion (42) in Corollary 2 can be reformulated as

$$\begin{array}{cc}\hfill {(-1)}^k{\left({\displaystyle \frac{2\mathrm{arccosh}t}{\pi }}\right)}^{2k}=& 1+\sum _{r=2}^{\infty }{(-1)}^r\left[\sum _{\ell =1}^r{(-1)}^{\ell }{\displaystyle \frac{{\langle 2k\rangle }_{2\ell }}{{\pi }^{2\ell }}}Q(2\ell ,2r-2\ell )\right]{\displaystyle \frac{{\left(2t\right)}^{2r}}{\left(2r\right)!}}\hfill \\ & +\sum _{r=1}^{\infty }{(-1)}^r\left[\sum _{\ell =1}^r{(-1)}^{\ell -1}{\displaystyle \frac{{\langle 2k\rangle }_{2\ell -1}}{{\pi }^{2\ell -1}}}Q(2\ell -1,2r-2\ell )\right]{\displaystyle \frac{{\left(2t\right)}^{2r-1}}{(2r-1)!}}\hfill \end{array}$$

for $k\in \mathbb{N}$ and $\left|t\right|<1$.

Remark 5. 

For $\left|t\right|<1$, the power series expansion (41) can be reformulated as

$${\left(\mathrm{arccosh}t\right)}^2=-{\displaystyle \frac{{\pi }^2}{4}}+\pi \sum _{r=1}^{\infty }{[(2r-3)!!]}^2{\displaystyle \frac{t^{2r-1}}{(2r-1)!}}-2\sum _{r=1}^{\infty }{[(2r-2)!!]}^2{\displaystyle \frac{t^{2r}}{\left(2r\right)!}}.$$

(57)

The series expansions (47) and (57) are more beautiful and concise in form.

Remark 6. 

The relations

$$\arccos t=2\arctan \sqrt{{\displaystyle \frac{1-t}{1+t}}},-1<t\le 1$$

and

$$\arcsin t=2\arctan {\displaystyle \frac{t}{1+\sqrt{1-t^2}}},\left|t\right|\le 1$$

can be utilized to derive series expansions of powers of the inverse tangent function $\arctan t$ from series expansions (11), (33), (38), and (47) of powers of the inverse cosine function $\arccos t$.

Remark 7. 

On infinite series representations (45) and (46) in Theorem 9 and the infinite series representation (51) in Corollary 3, we would like to mention the first unsolved problem posed by Herbert S. Wilf (1931–2012) on 13 December 2010 at https://www2.math.upenn.edu/~wilf/website/UnsolvedProblems.pdf (accessed on 18 July 2024).

Wilf’s third problem on the above website has been further studied in [34] [Section 5] and the preprint at the site https://doi.org/10.48550/arXiv.2110.08576 (accessed on 18 July 2024) has been further studied by the author of this paper.

Remark 8. 

In the identities (55) and (56) in Theorem 12, the central binomial coefficient $\left({\displaystyle \genfrac{}{}{0pt}{}{2r}{r}}\right)$ is involved several times. For more information on central binomial coefficient $\left({\displaystyle \genfrac{}{}{0pt}{}{2r}{r}}\right)$ and its extension or generalization, please refer to the paper [10] and the closely related references therein.

Remark 9. 

It is highly significant to simplify $Q(m,n)$, as performed in (39) and (40).

Remark 10. 

Formulas (15) and (16) in Theorem 4 can be rearranged as

$$\begin{array}{c}{B}_{n,n-2j}\left(1,0,1,0,9,0,225,0,\dots ,{[(2j-3)!!]}^2,0,{[(2j-1)!!]}^2\right)\hfill \\ =B_{n,n-2j}\left({\left(\arcsin t\right)}^{\prime }{{|}_{t=0},{\left(\arcsin t\right)}^{″}|}_{t=0},\dots ,{\left(\arcsin t\right)}^{(2j+1)}{|}_{t=0}\right)\hfill \\ ={(-1)}^{n-2j}{B}_{n,n-2j}\left({\left(\arccos t\right)}^{\prime }{{|}_{t=0},{\left(\arccos t\right)}^{″}|}_{t=0},\dots ,{\left(\arccos t\right)}^{(2j+1)}{|}_{t=0}\right)\hfill \\ ={(-1)}^j{2}^{2j}Q(n-2j,2j)\hfill \end{array}$$

(58)

and

$$\begin{array}{c}{B}_{n-1,n-2j}\left(1,0,1,0,9,0,225,0,\dots ,{[(2j-3)!!]}^2,0\right)\hfill \\ =B_{n-1,n-2j}\left({\left(\arcsin t\right)}^{\prime }{{|}_{t=0},{\left(\arcsin t\right)}^{″}|}_{t=0},\dots ,{\left(\arcsin t\right)}^{\left(2j\right)}{|}_{t=0}\right)\hfill \\ ={(-1)}^{n-2j}{B}_{n-1,n-2j}\left({\left(\arccos t\right)}^{\prime }{{|}_{t=0},{\left(\arccos t\right)}^{″}|}_{t=0},\dots ,{\left(\arccos t\right)}^{\left(2j\right)}{|}_{t=0}\right)\hfill \\ =0\hfill \end{array}$$

(59)

for $n-2j\ge 1$ and $j\in \mathbb{N}$, where $Q(k,2r)$ is given by (8).

Formulas (58), (59), and those formulas in Theorem 4 have been applied on the site https://math.stackexchange.com/a/4671660 (accessed on 3 April 2024) to obtain the Maclaurin power series expansion

$${\displaystyle \frac{{\mathrm{e}}^{a\arcsin t}}{\sqrt{1-t^2}}}=\sum _{n=0}^{\infty }{a}^n\left[1+\sum _{j=1}^{\lfloor n/2\rfloor }{(-1)}^j{\left({\displaystyle \frac{2}{a}}\right)}^{2j}\sum _{k=0}^{2j}\left({\displaystyle \genfrac{}{}{0pt}{}{n+k-2j}{n-2j}}\right)s(n,n+k-2j){\left({\displaystyle \frac{n-1}{2}}\right)}^k\right]{\displaystyle \frac{t^n}{n!}}$$

for $\left|t\right|<1$ by

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{{\mathrm{e}}^{a\arcsin t}}{\sqrt{1-t^2}}}\right)}^{\left(n\right)}& ={\displaystyle \frac{1}{a}}{\displaystyle \frac{{\mathrm{d}}^{n+1}}{\mathrm{d}{t}^{n+1}}}\left({\mathrm{e}}^{a\arcsin t}\right)\hfill \\ & ={\displaystyle \frac{1}{a}}\sum _{k=1}^{n+1}{\mathrm{e}}^{a\arcsin t}{B}_{n+1,k}\left({\left(a\arcsin t\right)}^{\prime },{\left(a\arcsin t\right)}^{″},\dots ,{\left(a\arcsin t\right)}^{n-k+2}\right)\hfill \\ & \to \sum _{k=1}^{n+1}{a}^{k-1}{B}_{n+1,k}\left({\left(\arcsin t\right)}^{\prime }{|}_{t=0},{\left(\arcsin t\right)}^{″}{{|}_{t=0},\dots ,{\left(\arcsin t\right)}^{n-k+2}|}_{t=0}\right)\hfill \\ & =\sum _{k=1}^{n+1}{a}^{k-1}{B}_{n+1,k}\left(1,0,1,0,9,0,225,0,\dots ,{\left(\arcsin t\right)}^{n-k+2}{|}_{t=0}\right)\hfill \\ & =\sum _{\ell =0}^n{a}^{n-\ell }{B}_{n+1,n+1-\ell }\left(1,0,1,0,9,0,225,0,\dots ,{\left(\arcsin t\right)}^{\ell +1}{|}_{t=0}\right)\hfill \\ & =a^n+\sum _{j=1}^{\lfloor n/2\rfloor }{a}^{n-2j}{B}_{n+1,n+1-2j}\left(1,0,1,0,9,0,225,0,\dots ,{\left(\arcsin t\right)}^{2j+1}{|}_{t=0}\right)\hfill \\ & =a^n+a^n\sum _{j=1}^{\lfloor n/2\rfloor }{(-1)}^j{\left({\displaystyle \frac{2}{a}}\right)}^{2j}\sum _{k=0}^{2j}\left({\displaystyle \genfrac{}{}{0pt}{}{n+k-2j}{n-2j}}\right)s(n,n+k-2j){\left({\displaystyle \frac{n-1}{2}}\right)}^k\hfill \end{array}$$

as $t\to 0$, where an empty sum is understood to be 0 and the notation $\lfloor t\rfloor$ denotes the floor function whose value is the largest integer less than or equal to $t\in \mathbb{R}$.

Additionally, we recommend the following sites for some answers to several questions related to Theorems 1 and 7 in this paper:

• https://math.stackexchange.com/a/4157803 (accessed on 12 April 2024);
• https://math.stackexchange.com/a/4379999 (accessed on 13 March 2023);
• https://math.stackexchange.com/a/4380027 (accessed on 13 March 2023);
• https://math.stackexchange.com/a/4657809 (accessed on 12 March 2023).

Remark 11. 

Theorem 3.1 in [35] reads that, if an analytic function $f\left(t\right)$ satisfies

$$f^j\left(t\right)=\sum _{n=0}^{\infty }{C}_{j,n}{\displaystyle \frac{t^n}{n!}},j\in {\mathbb{N}}_0=\{0,1,2,\dots \},$$

then

$$f^{\alpha }\left(t\right)=\sum _{n=0}^{\infty }\left[\sum _{k=0}^n{\displaystyle \frac{{(-\alpha )}_k}{k!}}\sum _{q=0}^k{(-1)}^q\left({\displaystyle \genfrac{}{}{0pt}{}{k}{q}}\right)f^{\alpha -q}\left(0\right)C_{q,n}\right]{\displaystyle \frac{t^n}{n!}},\alpha \in \mathbb{R}.$$

This conclusion is general, theoretic, and applicable.

Remark 12. 

On 4 August 2024, the author of this paper found the paper [36], in which, among other aspects, the authors Grishaev and Sazonov were influenced by the techniques, ideas, and thoughts of the papers [5,12,19,22,24,35,37], in the preprint at the site https://doi.org/10.21203/rs.3.rs-959177/v3 (accessed on 16 July 2024), and in Remark 11 above; they then established the Maclaurin power series expansions

$${\left({\displaystyle \frac{\arctan t}{t}}\right)}^m=m!\sum _{n=0}^{\infty }{(-1)}^{n}G(m,n)t^{2n},\left|t\right|<1$$

and

$${\left({\displaystyle \frac{\arctan t}{t}}\right)}^r=1+\sum _{m=1}^{\infty }{(-1)}^m\left[\sum _{k=1}^{2m}{\displaystyle \frac{{(-r)}_k}{k!}}\sum _{j=1}^k{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right)j!G(j,m)\right]t^{2m},\left|t\right|<1$$

(60)

for $m\in \mathbb{N}$ and $r\in \mathbb{R}$, where

$$G(m,n)=\sum _{k=0}^{2n}\left({\displaystyle \genfrac{}{}{0pt}{}{2n+m-1}{2n-k}}\right)2^k{\displaystyle \frac{s(k+m,m)}{(k+m)!}}$$

for $m\in \mathbb{N}$ and $n\in {\mathbb{N}}_0$.

We observe that the Maclaurin power series expansion (60) can be reformulated as

$${\left({\displaystyle \frac{\arctan t}{t}}\right)}^r=1+\sum _{m=1}^{\infty }{(-1)}^m\left[\sum _{k=1}^{2m}{(-r)}_k\sum _{j=1}^k{\displaystyle \frac{{(-1)}^j}{(k-j)!}}G(j,m)\right]t^{2m}$$

for $r\in \mathbb{R}$ and $\left|t\right|<1$.

Remark 13. 

In the papers [37,38], Qi and his coauthors established the following related Maclaurin power series expansions.

Theorem 14 

([37] [p. 102, Theorem 11]). When $r>0$, the series expansions

$${\left({\displaystyle \frac{\sin t}{t}}\right)}^r=\sum _{q=0}^{\infty }{(-1)}^q\left[\sum _{k=0}^{2q}{\displaystyle \frac{{(-r)}_k}{k!}}\sum _{j=0}^k{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right){\displaystyle \frac{T(2q+j,j)}{\left(\genfrac{}{}{0pt}{}{2q+j}{j}\right)}}\right]{\displaystyle \frac{{\left(2t\right)}^{2q}}{\left(2q\right)!}}$$

(61)

and

$${\left({\displaystyle \frac{\sin t}{t}}\right)}^r=1+\sum _{q=1}^{\infty }{(-1)}^q\left[\sum _{k=1}^{2q}{\displaystyle \frac{{(-r)}_k}{k!}}\sum _{j=1}^k{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right)\sum _{m=0}^{2q}{(-1)}^m\left({\displaystyle \genfrac{}{}{0pt}{}{2q}{m}}\right){\left({\displaystyle \frac{j}{2}}\right)}^m{\displaystyle \frac{S(2q+j-m,j)}{\left(\genfrac{}{}{0pt}{}{2q+j-m}{j}\right)}}\right]{\displaystyle \frac{{\left(2t\right)}^{2q}}{\left(2q\right)!}}$$

(62)

are convergent in $t\in \mathbb{C}$, where the rising factorial ${(-r)}_k$ is defined by (2), the Stirling numbers of the second kind $S(2q+j-m,j)$ are generated by (5) or (6), and the central factorial numbers of the second kind $T(2q+j,j)$ can be generated [39,40] by

$${\displaystyle \frac{1}{\ell !}}{\left(2\sinh {\displaystyle \frac{t}{2}}\right)}^{\ell }=\sum _{n=\ell }^{\infty }T(n,\ell ){\displaystyle \frac{t^n}{n!}},\ell \in {\mathbb{N}}_0$$

and can be computed [41] [Chapter 6, Equation (26)] by

$$T(n,\ell )={\displaystyle \frac{1}{\ell !}}\sum _{j=0}^{\ell }{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{\ell }{j}}\right){\left({\displaystyle \frac{\ell }{2}}-j\right)}^n$$

with $T(0,0)=1$ and $T(n,0)=0$ for $n\in \mathbb{N}$; see also [39] [Proposition 2.4, (xii)].

When $r<0$, the series expansions (61) and (62) are convergent in $\left|t\right|<\pi$.

Theorem 15 

([38] [p. 11]). If $r>0$, the Maclaurin power series expansion

$${\cos }^{r}t=\sum _{k=0}^{\infty }{(-1)}^k\left[\sum _{\ell =0}^{2k}{\displaystyle \frac{{(-r)}_{\ell }}{\ell !}}\sum _{m=0}^{\ell }{\displaystyle \frac{{(-1)}^m}{2^m}}\left({\displaystyle \genfrac{}{}{0pt}{}{\ell }{m}}\right)\sum _{q=0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{m}{q}}\right){(2q-m)}^{2k}\right]{\displaystyle \frac{t^{2k}}{\left(2k\right)!}}$$

(63)

is convergent for $t\in \mathbb{C}$; if $r<0$, the series expansion (63) is convergent for $\left|t\right|<{\displaystyle \frac{\pi }{2}}$.

Making use of the relation

$${\left({\displaystyle \frac{\tan t}{t}}\right)}^r={\left({\displaystyle \frac{\sin t}{t}}\right)}^r{\cos }^{-r}t,r\in \mathbb{R},$$

employing Theorems 14 and 15, and multiplying the Maclaurin power series expansions of ${\left({\displaystyle \frac{\sin t}{t}}\right)}^r$ and ${\cos }^{-r}t$, we can derive an explicit and closed-form expression of the Maclaurin power series expansion of the real power ${\left({\displaystyle \frac{\tan t}{t}}\right)}^r$ in $\left|t\right|<\pi$ for $r<0$ or $\left|t\right|<{\displaystyle \frac{\pi }{2}}$ for $r>0$. We believe that this is a new result.

Remark 14. 

This paper is a revised version of the preprint at the site https://doi.org/10.21203/rs.3.rs-959177/v3 (accessed on 16 July 2024) and a sibling of the articles [5,19,24,37].

8. Conclusions

The main question addressed in this paper is how to establish power series expansions for real powers of the inverse trigonometric and inverse hyperbolic functions, specifically the inverse cosine and inverse sine functions, as well as the inverse hyperbolic cosine and sine functions. Additionally, this paper explores how these expansions can be used to derive combinatorial identities involving the Stirling numbers of the first kind and central binomial coefficients, and to find infinite series representations of the circular constant $\pi$ and its real powers.

The originality lies in the approach taken to establish these power series expansions using the Faà di Bruno formula, explicit formulas for partial Bell polynomials, and combinatorial identities. The research is relevant as it provides new closed-form formulas for partial Bell polynomials and new infinite series representations of $\pi$ and its powers, contributing to both the fields of combinatorics and mathematical analysis. The specific gap addressed is the lack of explicit power series expansions for these specific functions and their connections to combinatorial identities and mathematical constants.

Compared to other published material, this paper provides a thorough investigation of power series expansions of real powers of inverse trigonometric and hyperbolic functions, incorporating combinatorial identities and explicit series representations of $\pi$. It offers a unique approach by linking these expansions to the Stirling numbers of the first kind and central binomial coefficients, extending the utility of these mathematical tools in various fields such as statistical mechanics and plasma physics. This paper also revisits and builds upon existing work, offering new insights and expanded applications.