The reducibility Of An Airy Operator

We show that the determinant $\mathrm{\nabla }(d,\alpha ),$ which seems to be not considered in the past, is not zero. As an application of this result we prove that the Setoyanagi operator $S_{p,q}={\mathrm{\partial }}^2-(a{x}^p+b{x}^q)$ is irreducible over $\mathbb{C}(x)[\mathrm{\partial }]$.