Recursive construction of higher-order alpha-derivations

Let $A\subseteq\mathbb{K}[x]$ be a subalgebra, let $\alpha$ be the evaluation point used in the definition of an $\alpha$-derivation, and let $L_1,L_3,\ldots,L_n$ be the recursively constructed linear maps from the coefficients $C_i^{(n)}$ and the operators $D_j$ described immediately before the claim. For odd $n$ define

$$L_n=\sum_{i=0}^n C_i^{(n)}c^iD_{n-i}.$$

Higher-order derivation conjecture. If

$$L_1(f)=L_3(f)=\cdots=L_{n-2}(f)=0$$

for every $f\in A$, then $L_n$ is an $\alpha$-derivation in $A$. The preceding theorem gives uniqueness of the coefficients $C_i^{(n)}$ satisfying the required relations; the conjecture asserts that the resulting next map has the derivation property under the stated vanishing conditions.