A combinatorial proof of an identity for the divisor generating function

TL;DR

This paper provides a combinatorial proof of Uchimura's identity for the divisor function and its generalizations, translating analytic identities into partition combinatorics and deriving related combinatorial results.

Contribution

It introduces a novel combinatorial proof for Uchimura's identity and its generalizations, connecting divisor functions with partition combinatorics.

Findings

Combinatorial proof of Uchimura's identity established

Generalizations of the identity are also proved combinatorially

Additional combinatorial results are derived as by-products

Abstract

We translate Uchimura's identity for the divisor function and whose generalizations into combinatorics of partitions, and give a combinatorial proof of them. As a by-product of their proofs, we obtain some combinatorial results.