On the Taylor coefficients of $\arcsin^3$

Question

The Maclaurin series of $\arcsin^2$ and $\arcsin^4$ are fairly well-known, $$ \arcsin^2(x) = \sum_{n\geq 1}\frac{(2x)^{2n}}{2n^2\binom{2n}{n}},\qquad \arcsin^4(x)=3\sum_{n\geq 1}\frac{H_{n-1}^{(2)}(2x)^{2n}}{2n^2\binom{2n}{n}} $$ but in order to deal with some logarithmic integrals I need the Maclaurin series of $\arcsin^3(x)$.

Mr. Wolfram states this is a result of Ramanujan, but I have not been able to find it in his notebooks, so I would like some help. Any derivation from scratch is clearly just as welcome.

Answer 1

See Ramanujan's Notebooks. Part 1 at page 263 For a more general result see also the paper "Integer Powers of Arcsin" by J. M. Borwein and M. Chamberland.

Answer 2

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 $m\ge2$. In particular, we have \begin{align} \frac{\arcsin x}{x}&=1!\sum_{n=0}^{\infty}[(2n-1)!!]^2\frac{x^{2n}}{(2n+1)!},\\ \biggl(\frac{\arcsin x}{x}\biggr)^2&=2!\sum_{n=0}^{\infty} [(2n)!!]^2 \frac{x^{2n}}{(2n+2)!},\\ \biggl(\frac{\arcsin x}{x}\biggr)^3 &=3!\sum_{n=0}^{\infty}[(2n+1)!!]^2 \Biggl[\sum_{k=0}^{n}\frac{1}{(2k+1)^2}\Biggr]\frac{x^{2n}}{(2n+3)!},\\ \biggl(\frac{\arcsin x}{x}\biggr)^4&=4!\sum_{n=0}^{\infty}[(2n+2)!!]^2\Biggl[\sum_{k=0}^{n}\frac{1}{(2k+2)^2}\Biggr] \frac{x^{2n}}{(2n+4)!},\\ \biggl(\frac{\arcsin x}{x}\biggr)^5&=\frac{5!}{2}\sum_{n=0}^{\infty}[(2n+3)!!]^2 \Biggl[\Biggl(\sum_{k=0}^{n+1}\frac{1}{(2k+1)^2}\Biggr)^2 -\sum_{k=0}^{n+1}\frac{1}{(2k+1)^4}\Biggr] \frac{x^{2n}}{(2n+5)!}. \end{align} These results have been reviewed and surveyed in the papers [2, 3] 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.

Answer 3

A more general conclusion is as follows.

For $\alpha\in\mathbb{R}$ and $|z|<1$, we have \begin{equation}\label{arcsin-real-power-EQ} \biggl(\frac{\arcsin z}{z}\biggr)^\alpha =1+\sum_{n=1}^{\infty} (-1)^{n}\Biggl[\sum_{k=1}^{2n} \frac{(-\alpha)_k}{(2n+k)!}\sum_{q=1}^{k}(-1)^{q}\binom{2n+k}{k-q}Q(q,2n)\Biggr](2z)^{2n}, \end{equation} where $(z)_k$ is defined by \begin{equation}\label{rising-Factorial} (z)_n=\prod_{k=0}^{n-1}(z+k) = \begin{cases} z(z+1)\dotsm(z+n-1), & n\in\mathbb{N}\\ 1, & n=0 \end{cases} \end{equation} and $Q(q,2n)$ is given by \begin{equation*} Q(m,n)=\sum_{k=0}^{n} \binom{m+k-1}{m-1} s(m+n-1,m+k-1)\biggl(\frac{m+n-2}{2}\biggr)^{k} \end{equation*} for $m\in\mathbb{N}$ and $n\ge2$ and the Stirling numbers of the first kind $s(n,k)$ can be analytically computed by \begin{equation}\label{1st-stirling-reciprocal-formula} |s(n+1,k+1)|=n!\sum_{\ell_1=k}^{n} \frac1{\ell_1}\sum_{\ell_2=k-1}^{\ell_1-1}\frac1{\ell_2}\dotsm \sum_{\ell_{k-1}=2}^{\ell_{k-2}-1} \frac1{\ell_{k-1}} \sum_{\ell_{k}=1}^{\ell_{k-1}-1}\frac1{\ell_{k}}, \quad n\ge k\in\mathbb{N}. \end{equation} See Theorem 7 in the reference [1] and Example 3.1 in the reference [2] below.

References

1. F. Qi, 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, Symmetry 16 (2024), no. 9, Article 1145, 21 pages; available online at https://doi.org/10.3390/sym16091145.
2. F. Qi, G. V. Milovanovic, and D. Lim, Specific values of partial Bell polynomials and series expansions for real powers of functions and for composite functions, Filomat 37 (2023), no. 28, 9469--9485; available online at https://doi.org/10.2298/FIL2328469Q.