Coefficients of prolongations for symmetries of ODEs

Sophus Lie developed a systematic way to solve ODEs. He found that transformations which form a continuous group and leave a differential equation invariant can be used to simplify the equation. Lie’s method uses the infinitesimal generator of these point transformations. These are symmetries of the equation mapping solutions into solutions. Lie’s methods did not find widespread use in part because the calculations for the infinitesimals were quite lengthy, needing to calculate the prolongations of the infinitesimal generator. Nowadays, prolongations are obtained using Maple or Mathematica, and Lie’s theory has come back to the attention of researchers. In general, the computation of the coefficients of the (n)prolongation is done using recursion formulas. Others have given methods that do not require recursion but use Fréchet derivatives. In this paper, we present a combinatorial approach to explicitly write the coefficients of the prolongations. Besides being novel, this approach was found to be useful by the authors for didactical and combinatorial purposes, as we show in the examples.