Note on expanding implicit functions into formal power series by means of multivariable Stirling polynomials

TL;DR

This paper develops a formal power series expansion for implicit functions of two variables using multivariable Stirling polynomials, Bell polynomials, and their orthogonal counterparts.

Contribution

It introduces a novel method to derive power series for implicit functions based solely on the original function's Taylor coefficients, utilizing advanced combinatorial polynomials.

Findings

Derived a formal series for implicit functions with coefficients depending only on original Taylor coefficients.

Utilized partial Bell polynomials and their orthogonal companions in the derivation.

Provided a systematic approach for expanding implicit functions into formal power series.

Abstract

Starting from the representation of a function $f(x,y)$ as a formal power series with Taylor coefficients $f_{m,n}$, we establish a formal series for the implicit function $y=y(x)$ such 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 orthogonal companions.