Some nonlinear inverse relations of Bell polynomials via the Lagrange inversion formula

TL;DR

This paper uses the Lagrange inversion formula to establish new nonlinear inverse relations for Bell polynomials, providing simplified proofs and convolution identities, advancing combinatorial and polynomial theory.

Contribution

It introduces a general nonlinear inverse relation for Bell polynomials using the Lagrange inversion formula, extending previous results and deriving new identities.

Findings

New nonlinear inverse relations for Bell polynomials

Simplified proof of existing inverse relations

Convolution identities for Mina polynomials

Abstract

In this paper, by means of the classical Lagrange inversion formula, we establish a general nonlinear inverse relations which is a partial solution to the problem proposed in the paper [J. Wang, Nonlinear inverse relations for the Bell polynomials via the Lagrange inversion formula, J. Integer Seq., Vol. 22 (2019), Article 19.3.8. (https://cs.uwaterloo.ca/journals/JIS/VOL22/Wang/wang53.pdf). As applications of this inverse relation, we not only find a short proof of another nonlinear inverse relation due to Birmajer et al., but also set up a few convolution identities concerning the Mina polynomials.