Computing the hypergometric function $_1F_2$ in special cases I need to compute integrals involving spherical Bessel functions, which led me to compute the following hypergeometric function :
$_1F_2((\alpha+1)/2;\beta+3/2,(\alpha+3)/2;z)$ 
Also, in my particular problem : 
$\bullet$  $\alpha$ and $\beta$ are always integers
$\bullet$ $\alpha \geq \beta \geq 2$ 
$\bullet$  $z$ is always real
$\bullet$ $z \leq 0$ and can be anywhere between 0 and $\sim -10^{13}$
I have tried to use a couple of math libraries (Arb in C, mpmath in python) that implement the $_1F_2$ function. Unfortunately, I always end up with the same problem : for very large values of $|z|$, the computation fails. I assume this is due to convergence issues, as the code is trying to brute force the computation (I think) instead of using asymptotic approximations. Arb has those approximations, but only for other hypergeometric functions, such as $_0F_1$. Also, for mpmath the computation time also becomes an issue for large $|z|$.
Since my mathematical skills are quite limited, my question is to know whether it is possible (given the specific parameters I'm using) to simplify the $_1F_2$ function or maybe express it with other functions (hypergeometric or not) whose computation is more robust in the libraries I mention ?
Thanks !
Edit :
It might be important to mention that the function I'm trying to evaluate is actually $_1F_2((\alpha+1)/2;\beta+3/2,(\alpha+3)/2;-X^2)$, with $X$ being a real between 0 and $\sim 10^6 - 10^7$
 A: We have
$$\begin{eqnarray*}\phantom{}_1 F_2\left(\frac{a+1}{2};b+\frac{3}{2},\frac{a+3}{2}; z\right)=\sum_{n\geq 0}\frac{(a+1)}{(a+1+2n)}\cdot\frac{\left(b+\frac{1}{2}\right)!}{\left(b+n+\frac{1}{2}\right)!}\cdot \frac{z^n}{n!}\end{eqnarray*}$$
$$\begin{eqnarray*}\phantom{}_1 F_2\left(\frac{a+1}{2};b+\frac{3}{2},\frac{a+3}{2}; -z^2\right)&=&\sum_{n\geq 0}\frac{(a+1)}{(a+1+2n)}\cdot\frac{\left(b+\frac{1}{2}\right)!}{\left(b+n+\frac{1}{2}\right)!}\cdot \frac{(-1)^n z^{2n}}{n!}\\[0.2cm]&=&\frac{a+1}{z^{a+1}}\int_{0}^{z}x^a\,\phantom{}_0 F_1\left(;\frac{3}{2}+b;-x^2\right)\,dx\end{eqnarray*}$$
where 
$g_b(x)=\phantom{}_0 F_1\left(;\frac{3}{2}+b;-x^2\right)$, for small values of $b\in\mathbb{N}$, is given by
$$ g_0(x)=\frac{\sin(2x)}{2x},\quad g_1(x)=\frac{2\sin(2x)-6x\cos(2x)}{8x^3},\\ g_2(x)=\frac{15(3-4x^2)\sin(2x)-90x\cos(2x)}{32x^5} $$
For a specific value of $b$ it is not difficult (by induction) to compute the closed form of $g_b(x)$, then apply integration by parts to compute the closed form of the wanted hypergeometric function.
