Recursive Prolongation Formula

The recursive prolongation formula is the fundamental algorithm used to construct the vector fields $X_1,\dots ,X_{m-1}$ that constitute a canonical cinf-structure.

Unlike the classical prolongation formula for lambda-symmetrys which depends on a single function $\lambda$, this formula generates the $k$-th order component of the $i$-th vector field, denoted as ${\eta }_i^k$, by iterating on the previous order components and incorporating the previous vector field in the sequence.

The Formula

For a canonical structure defined by the smooth functions ${\lambda }_1,\dots ,{\lambda }_{m-1}$, the components ${\eta }_i^k$ (coefficient of ${\partial }_{u_k}$ for the vector field $X_i$) are determined recursively by:

Where:

• $A$ is the total derivative operator associated with the ODE.
• ${\lambda }_i$ is the $i$-th structural function.
• ${\eta }_{i-1}^{k-1}$ is the component from the previous vector field in the structure (with ${\eta }_0^k=0$).

Initial Conditions

The recursion is initialized based on the triangular nature of the canonical basis:

• ${\eta }_i^{i-1}=1$
• ${\eta }_1^k=(A+{\lambda }_1)({\eta }_1^{k-1})$ for all $k=1,\dots ,m-1$

Remarks

• Diagonal Components: As a consequence of the recursion, the diagonal components sum the structural functions:$${\eta }_i^i=\sum _{j=1}^i{\lambda }_j$$
• Determining Equations: This explicit construction allows one to substitute the ${\eta }_i^k$ expressions directly into the determining equations for cinf-structures, transforming the problem into a search for the scalar functions ${\lambda }_i$.