What question did this study set out to answer?

This work aims to expand implicit functions into formal power series using partial Bell polynomials.

May 10, 2026Open Access

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

Puntos clave

This work aims to expand implicit functions into formal power series using partial Bell polynomials.
Develops a formal series for the implicit function y = y(x) based on Taylor coefficients, fm,n.
Utilizes partial Bell polynomials and their inverse metaphors to determine the series coefficients.
Discusses various examples and applications of the formulated power series.
Establishes a method for representing implicit functions using power series derived from Taylor coefficients.
Demonstrates the applicability of partial Bell polynomials in expanding complex functions.

Resumen

Starting from the representation of a function f(x, y) as a formal power series with Taylor coefficients fm,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 fm,n. The solution to this problem provided here relies on using partial Bell polynomials and their inverse companions. Some examples and applications are discussed.

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