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

2023/07/05 by Schreiber, Alfred
#11B83 #11C08 #Combinatorics (math.CO) #FOS: Mathematics #Primary: 13F25 #Secondary: 05A19

paper · doi:10.48550/arxiv.2307.02638

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.

Related

• Explicit Implicit Function Theorem for All Fields
• Exponential Taylor Series
• De Moivre and Bell polynomials
• Power series expansion of Wilf function
• Summation of Divergent Power Series by Means of Factorial Series