I would like to obtain an integral representation for $2\operatorname{Ai}^2(z)\operatorname{Ai}'(z)$. Is it permissible to interchange differentiation and integration on the known formula$$\operatorname{Ai}^3(z)=\frac{1}{\left(72 \pi^5\right)^{1 / 2} i} \int_{\mathscr{L}_1} t^{1 / 2} K_{1 / 6}(T) \exp \left(\frac{5}{27} t^3-z t\right) d t$$to obtain
$$\operatorname{2Ai^2(z)Ai'(z)} = \frac{-1}{\left(72 \pi^5\right)^{1 / 2} i} \int_{\mathscr{L}_1} t^{3 / 2} K_{1 / 6}(\frac{4}{27} t^3) \exp \left(\frac{5}{27} t^3-z t\right) d t?$$
where $K_{1/6}$ is the modified bessel function of the second kind of order $1/6$.
