I have the following Hypergeometric function of the first kind:
$_{1}F_1(\frac{1}{2}, \frac{3}{2}, -x)$
where $x$ is not negative. This function can also be written as the following series:
$\sum_{n=0}^{\infty} \frac{1}{n!} \,\frac{(-1)^n}{2n+1} x^n = 1-\frac{1}{3}x + \frac{1}{5} \, \frac{x^2}{2!} - \frac{1}{7} \, \frac{x^3}{3!} + ...$
My question is: Can this special case of the Hypergeometric function be expressed in any other closed form (or another function)?
Thanks
As commented by GEdgar, set $x=y^2$. So,$$S=\sum_{n=0}^{\infty} \frac{1}{n!} \,\frac{(-1)^n}{2n+1} x^n =\sum_{n=0}^{\infty} \frac{1}{n!} \,\frac{(-1)^n}{2n+1} y^{2n}=\frac 1 y\sum_{n=0}^{\infty} \frac{1}{n!} \,\frac{(-1)^n}{2n+1} y^{2n+1}$$ Now, cross multiply by $y$ and differentiate $$\frac {d}{dy}(y\,S)=\sum_{n=0}^{\infty} \frac{(-1)^n}{n!} \, y^{2n}=e^{-y^2}$$ Now, integrate $$y\,S=\int e^{-y^2}\,dy=\frac{\sqrt{\pi }}{2} \text{erf}(y)$$ So $$S=\frac{\sqrt{\pi }}{2}\, \frac{ \text{erf}(y)} y=\frac{\sqrt{\pi }}{2}\, \frac{ \text{erf}(\sqrt x)} {\sqrt x}$$