Jet-polynomial extraction

Let $X_{I,\gamma }$, $|\gamma |\le 2$, be ordinary derivative jets:

They are not factorial-normalized. Mixed derivatives are represented by their common multiindex; the jet derivations commute:

At a fixed $x$, construct ${{\mathcal{J}}_{\mathrm{D}}}^{\rho \sigma }(x;X)$ from the displayed finite formulas, treating ${\sigma }_{\epsilon }(x)$ and its derivatives as external coefficients. Apply (eq:source-conversion) to obtain ${{\mathcal{J}}_x}^L(X)$.