New identities for the partial Bell polynomials

TL;DR

This paper introduces a new explicit formula, addition formula, and recurrence relation for multivariate partial Bell polynomials, enabling easier direct evaluation and deeper understanding of their properties.

Contribution

It provides the first explicit closed-form expression, an addition formula, and a recurrence relation for multivariate partial Bell polynomials.

Findings

New explicit formula for $B_{n,k}$ involving multiple summations.

Addition formula with respect to $k$ for $B_{n,k}$.

New recurrence relation for the polynomials.

Abstract

A new explicit closed-form formula for the multivariate $(n, k)$th partial Bell polynomial $B_{n,k} (x_1, x_2, ..., x_{n - k + 1})$ is deduced. The formula involves multiple summations and makes it possible, for the first time, to easily evaluate $B_{n,k}$ directly for given values of $n$ and $k$ ($n\geq k, k =2, 3,...$). Also, a new addition formula (with respect to $k$) is found for the polynomials $B_{n,k}$ and it is shown that they admit a new recurrence relation. Several special cases and consequences are pointed out, and some examples are also given.