**Part I.** The following two formulas
\begin{equation}
{}_1F_2\biggl(1; n+1, n+\frac{3}{2}; -\frac{x^2}{4}\biggr)
=(-1)^n\frac{(2n+1)!}{x^{2n+1}} \Biggl[\sin x-\sum_{k=0}^{n-1} (-1)^k\frac{x^{2k+1}}{(2k+1)!}\Biggr]
\end{equation}
and
\begin{equation}
{}_1F_2\biggl(1;n+\frac{1}{2}, n+1; -\frac{x^2}{4}\biggr)
=(-1)^n\frac{(2n)!}{x^{2n}} \Biggl[\cos x-\sum_{k=0}^{n-1} (-1)^k\frac{x^{2k}}{(2k)!}\Biggr]
\end{equation}
for $n\in\mathbb{N}$ and $x>0$ were published on Page 16 of the reference [1] below. On the other hand, we know
\begin{equation}
{}_1F_2\biggl(1;1,\frac{3}{2}; -\frac{x^2}{4}\biggr)
=\frac{4 \operatorname{arcsinh}\bigl(\frac{x}{2}\bigr)}{x \sqrt{x^2+4}\,}
\end{equation}
and
\begin{equation}
{}_1F_2\biggl(1;\frac{1}{2},1; -\frac{x^2}{4}\biggr)
=\frac{2}{\sqrt{x^2+4}\,}.
\end{equation}
These four formulas reveal that the hypergeometric functions $$
{}_1F_2\biggl(1;n,n+\frac12;-x^2\biggr)\quad\text{and}\quad {}_1F_2\biggl(1;n-\frac12,n;-x^2\biggr)
$$
for $n\in\mathbb{N}$ and $x>0$ are elementary! In other words, the hypergeometric function
$$
{}_1F_2\biggl(1;\frac{n}2,\frac{n+1}2;-x^2\biggr), \quad n\in\mathbb{N}
$$
has a closed-form expression.

**Part II.** Among other things, combining the above first two formulas with Theorems 1 and 2 in the reference [1] below, we can conclude that both of the hypergeometric function ${}_1F_2\bigl(1; n+1, n+\frac{3}{2}; -\frac{x^2}{4}\bigr)$ for $n\ge1$ and the hypergeometric function ${}_1F_2\bigl(1;n+\frac{1}{2}, n+1; -\frac{x^2}{4}\bigr)$ for $n\ge2$ are positive and decreasing in $x\in(0,\infty)$, while both of them are concave in $x\in(0,\pi)$. Summing up, the hypergeometric function
$$
{}_1F_2\biggl(1; \frac{n+3}{2},\frac{n+4}{2}; -\frac{x^2}{4}\biggr), \quad n\in\mathbb{N}
$$
is positive and decreasing in $x\in(0,\infty)$, while it is concave in $x\in(0,\pi)$.

**Part III.** Considering the first two formulas in **Part I** above, we observe that, in the reference [2] below, among other things, the function
\begin{equation*}
\ln\biggl[{}_1F_2\biggl(1;n+\frac{1}{2}, n+1; -\frac{x^2}{4}\biggr)\biggr], \quad n\in\mathbb{N}
\end{equation*}
was expanded into a Maclaurin power series at the point $x=0$, as well as the function
\begin{equation*}
\frac{\ln\bigl[{}_1F_2\bigl(1;\frac{5}{2}, 3; -\frac{x^2}{4}\bigr)\bigr]}{\ln\cos x}
\end{equation*}
was proved to be decreasing on $\bigl(0,\frac{\pi}2\bigr)$.

For more information, please refer to Remark 7 in the paper [3] below.

References
 1. Tao Zhang, Zhen-Hang Yang, Feng Qi, and Wei-Shih Du, *Some properties of normalized tails of Maclaurin power series expansions of sine and cosine*, Fractal and Fractional **8** (2024), no. 5, Article 257, 17 pages; available online at https://doi.org/10.3390/fractalfract8050257.
 2. A. Wan and F. Qi, *Power series expansion, decreasing property, and concavity related to logarithm of normalized tail of power series expansion of cosine*, Electron. Res. Arch. **32** (2024), no. 5, 3130--3144; available online at https://doi.org/10.3934/era.2024143.
 3. Yue-Wu Li and Feng Qi, *A new closed-form formula of the Gauss hypergeometric function at specific arguments*, Axioms **13** (2024), no. 5, 24 pages; available online at https://www.researchgate.net/publication/380431446.