Another Identity for Complete Bell Polynomials based on Ramanujan's Congruences

TL;DR

This paper introduces a novel identity for complete Bell polynomials derived from Ramanujan's generating function for specific partition numbers, enriching the mathematical understanding of partition functions and Bell polynomials.

Contribution

The paper presents a new identity linking complete Bell polynomials to Ramanujan's congruences for partition numbers, offering a fresh perspective in combinatorial analysis.

Findings

Derived a new identity for complete Bell polynomials

Connected Bell polynomials with Ramanujan's partition congruences

Enhanced understanding of partition functions through Bell polynomial identities

Abstract

Let $p(n)$ be the number of partition of a positive integer $n$. We derive a new identity for complete Bell polynomials based on a generating function of $p(7n+5)$ given by Ramanujan.