 Formal pseudodifferential wave function

Treat $k$ as a formal spectral parameter near infinity. If $P={\sum }_{j\ge 0}{g}_j{\partial }^{\alpha -j}$, the formal pseudodifferential composition rule gives $Pw=k^{\alpha +\beta }{e}^{kx}{\sum }_{s\ge 0}{k}^{-s}{\sum }_{i+j+r=s}{g}_j\left(\genfrac{}{}{0ex}{}{\alpha -j}{r}\right)w_i^{(r)}$. Each coefficient contains finitely many terms. This describes a formal module action, not a claim that the infinite series converges at finite $k$.