Identities for partial Bell polynomials derived from identities for weighted integer compositions

TL;DR

This paper derives new identities for partial Bell polynomials by linking them to weighted integer compositions, providing elegant probabilistic proofs and extending the theoretical framework.

Contribution

It establishes that partial Bell polynomials are special cases of weighted integer compositions and derives new identities with probabilistic proofs.

Findings

Partial Bell polynomials are special cases of weighted integer compositions.

New identities for partial Bell polynomials are derived from general identities.

Probabilistic proofs using sums of discrete random variables are provided.

Abstract

We discuss closed-form formulas for the (n; k)-th partial Bell polynomials derived in Cvijovic. We show that partial Bell polynomials are special cases of weighted integer compositions, and demonstrate how the identities for partial Bell polynomials easily follow from more general identities for weighted integer compositions. We also provide short and elegant probabilistic proofs of the latter, in terms of sums of discrete integer-valued random variables. Finally, we outline further identities for the partial Bell polynomials.