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

Source 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.Accepted: 15 March 2026