Some convolution identities and an inverse relation involving partial Bell polynomials

TL;DR

This paper establishes new convolution formulas and an inverse relation for partial Bell polynomials, using a multinomial approach to derive combinatorial identities and explore their compositions.

Contribution

It introduces a novel inverse relation and convolution identities for partial Bell polynomials, expanding the combinatorial toolkit with a multinomial coefficient approach.

Findings

Derived an inverse relation for partial Bell polynomials

Established new convolution formulas involving Bell polynomials

Discussed known and new combinatorial identities of convolution type

Abstract

We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting multinomial formula for the binomial coefficients. The inverse relation is deduced from a parametrization of suitable identities that facilitate dealing with compositions of Bell polynomials.