Taylor expansion for $\arcsin^2{x}$

Question

I stumbled upon this particular expansion that was included in this post.

---

$$ \displaystyle \arcsin^{2}(x) = \frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{n^{2} \binom{2n}{n}} (2x)^{2n}$$

---

This caught my eye because I remember trying to derive a Taylor series for $\arcsin^{2}(x)$ a while ago without much success.

Can anyone prove this or point me to a material that would show a proof of this identity?

EDIT :

Feel free to use any mathematical apparatus at hand. I'm not interested in a proof fit for a certain level, nor am I looking for utmost elegance (though that would be lovely).

Answer 1

You can find a derivation for the Taylor series of $\frac{\arcsin(x)}{\sqrt{1-x^2}}$ in this nice answer. Since $$\frac{d\arcsin^2(x)}{dx} = \frac{2\arcsin(x)}{\sqrt{1-x^2}}$$

the Taylor series for $\arcsin^2(x)$ follows by integration.

Answer 2

There exists a general conclusion:

For $k\in\mathbb{N}$ and $|x|<1$, the function $\bigl(\frac{\arcsin x}{x}\bigr)^{k}$, whose value at $x=0$ is defined to be $1$, has Maclaurin's series expansion \begin{equation}\label{arcsin-series-expansion-unify} \biggl(\frac{\arcsin x}{x}\biggr)^{k} =1+\sum_{m=1}^{\infty} (-1)^m\frac{Q(k,2m)}{\binom{k+2m}{k}}\frac{(2x)^{2m}}{(2m)!}, \end{equation} where \begin{equation}\label{Q(m-k)-sum-dfn} Q(k,m)=\sum_{\ell=0}^{m} \binom{k+\ell-1}{k-1} s(k+m-1,k+\ell-1)\biggl(\frac{k+m-2}{2}\biggr)^{\ell} \end{equation} for $k\in\mathbb{N}$ and $m\ge2$.

This general result and its special cases such as $k=1,2,3,4,5$ have been established, reviewed, and surveyed in the papers [2, 3, 5] below, recovered in the paper [1] below, and generalized in the paper [4] below.

References

1. Feng Qi, Taylor's series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of Pi, Demonstratio Mathematica 55 (2022), no. 1, 710--736; available online at https://doi.org/10.1515/dema-2022-0157.
2. Bai-Ni Guo, Dongkyu Lim, and Feng Qi, Maclaurin's series expansions for positive integer powers of inverse (hyperbolic) sine and tangent functions, closed-form formula of specific partial Bell polynomials, and series representation of generalized logsine function, Applicable Analysis and Discrete Mathematics 16 (2022), no. 2, 427--466; available online at https://doi.org/10.2298/AADM210401017G.
3. Bai-Ni Guo, Dongkyu Lim, and Feng Qi, Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions, AIMS Mathematics 6 (2021), no. 7, 7494--7517; available online at https://doi.org/10.3934/math.2021438.
4. F. Qi, Explicit formulas for partial Bell polynomials, Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine and sine, and series representations of powers of Pi, Research Square (2021), available online at https://doi.org/10.21203/rs.3.rs-959177/v3.
5. https://math.stackexchange.com/a/4657809.

Answer 3

Starting from the well-known power series of $\arcsin$, \begin{equation} \newcommand{\where}[1]{\qquad(#1)} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\bigparens}[1]{\bigl(#1\bigr)} \newcommand{\N}{\mathbb{N}} \arcsin(x) = \sum_{n=0}^\infty \binom{2n}{n}\frac{x^{2n+1}}{4^n(2n+1)} =: \sum_{n=0}^\infty a_n x^{2n+1} \where{\abs{x}\le 1}, \end{equation} implies as a Cauchy product of two power series \begin{equation} \bigparens{\arcsin(x)}^2 = \sum_{n=0}^\infty b_n x^{2n+2} \where{\abs{x}\le 1}, \end{equation} with \begin{equation} b_n := \sum_{k=0}^n a_ka_{n-k} = 2^{2n+1}\frac{(n!)^2}{(2n+2)!} \where{n\in\N_0}, \end{equation} as can be shown by induction (or looked up in a table of series, e.g., eq. 1.645 2 in Gradshteyn).