Expansion of implicit functions into formal power series in terms of partial Bell polynomials

Alfred   Schreiber

doi:10.13069/jacodesmath.v13i2.397

Alfred Schreiber Department of Mathematics and Mathematical Education, Europa-Universit\"at Flensburg, EUF, Auf dem Campus 1, 24943~Flensburg, Germany

DOI: https://doi.org/10.13069/jacodesmath.v13i2.397

Keywords: Implicit function, Formal power series, Higher derivatives, Series reversion, Bell polynomials, Multivariable Stirling polynomials

Abstract

Starting from the representation of a function f(x, y) as a formal power series with Taylor coefficients f_m,n, a formal series is set up for the implicit function y = y(x) so that f(x, y) = 0 and the coefficients of the series for y depend exclusively on the f_m,n. The solution to this problem provided here relies on using partial Bell polynomials and their inverse companions. Some examples and applications are discussed.