Relationship between Bessel J and hypergeometric 0F1

Theorem

The following formula holds: $$J_{\nu}(z) = \left( \dfrac{z}{2} \right)^{\nu} \dfrac{1}{\Gamma(\nu+1)} {}_0F_1 \left(-;\nu+1;-\dfrac{z^2}{4} \right),$$ where $J_{\nu}$ denotes the Bessel function of the first kind, $\Gamma$ denotes the gamma function and ${}_0F_1$ denotes the hypergeometric 0F1.