📘 special functions

1. **State the problem:** We are given the function $$G_{q,\nu,r}[a,t] = \sum_{j=0}^\infty \frac{\Gamma(1-r)(-a)^j t^{(r+j)q-\nu-1}}{\Gamma(1+j)\Gamma(1-j-r)\Gamma((r+j)q-\nu)},$$

1. The problem asks us to show that the Bessel functions of the first kind for order $\frac{1}{2}$ and $-\frac{1}{2}$ can be expressed as $$J_{\frac{1}{2}}(x) = \sqrt{\frac{2}{\pi