power series of $\left(\frac{\sin z}z\right)^2$?

Question

So I need to find the power series of $\left(\frac{\sin z}z\right)^2$ around $0$, and find its radius of convergence.
Also the function is defined to be $1$ in $z=0$. My guess about the radius is infinity, but I'm stuck in the power series. This is what I've done so far:

$$\left(\frac{\sin z}z\right)^2 = \left(\frac{z-\frac{z^3}{3!}+\frac{z^5}{5!}-\cdots\frac{z^{2n+1}}{(2n+1)!}+\cdots}z\right)^2$$ $$=\left(1-\frac{z^2}{3!}+\frac{z^4}{5!}-\cdots\frac{z^{2n}}{(2n+1)!}+\cdots\right)^2$$ $$=\left(\sum_{n=0}^{\infty}\frac{(-1)^nz^{2n}}{(2n+1)!}\right)^2$$

could really use some help. Thanks very much!

Answer 1

Using Lord Shark the Unknown's hint we can rewrite the problem in the following way

$$\left(\frac{\sin z}z\right)^2=\frac1{2z^2}(1-\cos 2z)$$

Utilizing the well known Taylor Series of the cosine function this further becomes

$$\begin{align} \frac1{2z^2}(1-\cos 2z)&=\frac1{2z^2}\left(1-\sum_{n=0}^{\infty}(-1)^n\frac{(2z)^{2n}}{(2n)!}\right)\\ &=\frac1{2z^2}-2\sum_{n=0}^{\infty}(-1)^n\frac{(2z)^{2n-2}}{(2n)!}\\ &=-2\sum_{n=1}^{\infty}(-1)^n\frac{(2z)^{2n-2}}{(2n)!} \end{align}$$

$$\left(\frac{\sin z}z\right)^2=2\sum_{n=0}^{\infty}(-1)^n\frac{4^n z^{2n}}{(2n+2)!}$$

Regarding the radius of convergence one could apply the ratio test to verfity that it is actually infinity.

Answer 2

For $\ell\in\mathbb{N}_0$ and $z\in\mathbb{C}$, we have \begin{equation}\label{sine-power-ser-expan-eq} \biggl(\frac{\sin z}{z}\biggr)^{\ell} =1+\sum_{j=1}^{\infty} (-1)^{j}\frac{T(\ell+2j,\ell)}{\binom{\ell+2j}{\ell}} \frac{(2z)^{2j}}{(2j)!}, \end{equation} where \begin{equation}\label{T(n-k)-EF-Eq(3.1)} T(n,\ell)=\frac1{\ell!} \sum_{j=0}^{\ell}(-1)^{j}\binom{\ell}{j}\biggl(\frac{\ell}2-j\biggr)^n \end{equation} with $T(0,0)=1$ and $T(n,0)=0$ for $n\in\mathbb{N}$.

Reference

1. Feng Qi and Peter Taylor, Several series expansions for real powers and several formulas for partial Bell polynomials of sinc and sinhc functions in terms of central factorial and Stirling numbers of second kind, arXiv preprint (2022), available online at https://arxiv.org/abs/2204.05612v4 or https://doi.org/10.48550/arXiv.2204.05612.