Four explicit formulas for the prolongations of an infinitesimal Lie symmetry and multivariate Faa di Bruno formulas

Abstract

In 1979, building on S. Lie's theory of symmetries of (partial) differrential equations, P.J. Olver formulated inductive formulas which are appropriate for the computation of the prolongations of an infinitesimal Lie symmetry to jet spaces, for an arbitrary number n\geq 1 of independent variables (x^1, ..., x^n) and for an arbitrary number m \geq 1 of dependent variables (y^1, ..., y^m). This paper is devoted to elaborate a formalism based on multiple Kronecker symbols which enables one to handle these ``unmanageable'' prolongations and to discover the underlying complicated combinatorics. Proceeding progressively, we write down closed explicit formulas in four cases: n=m=1; n\geq 1, m=1; n=1, m\geq 1; general case n\geq 1, m\geq 1. As a subpart of the obtained formulas, we recover four possible versions of the (multivariate) Faà di Bruno formula. We do not employ the classical formalism based on the symmetric algebra ({\it cf.} e.g. H. Federer's book, p. 222), because it hides several explicit sums in symbolic compactifications, and because the presence of supplementary complexities ({\it e.g.} splitting of indices, combinatorics of partial derivatives) impedes us to apply such compactifications coherently. Our method of exposition is inductive: we conduct our reasonings by analyzing several thoroughly organized formulas, by comparing them together and by ``drifting'' towards generality, in homology with the classical style of L. Euler.