On a generalization of the Pentagonal Number Theorem

TL;DR

This paper generalizes the Pentagonal Number Theorem, deriving new identities and relations for infinite series, partitions, divisor sums, and Bell polynomials, expanding the theorem's applications.

Contribution

It introduces a broad generalization of the classical theorem, providing novel identities and recurrence relations for various mathematical functions.

Findings

New identities for infinite series and partitions

Recurrence relations for divisor sums

Identities involving Bell polynomials

Abstract

We study a generalization of the classical Pentagonal Number Theorem and its applications. We derive new identities for certain infinite series, recurrence relations and convolution sums for certain restricted partitions and divisor sums. We also derive new identities for Bell polynomials.