Identities and transformations for Lambert series and double Lambert series

TL;DR

This paper proves two new identities for Lambert series and double Lambert series, resolving several longstanding conjectures using classical transformation techniques.

Contribution

It introduces two identities for Lambert series and double Lambert series, solving conjectures by Andrews, Dixit, Schultz, Yee, Amdeberhan, and Ballantine.

Findings

Resolved conjectures on Lambert series identities.

Developed systematic rearrangement methods for double Lambert series.

Enhanced understanding of classical transformations in infinite series.

Abstract

We establish two identities for Lambert series and double Lambert series, thereby resolving conjectures of Andrews, Dixit, Schultz and Yee (Acta Arith.~181:253--286, 2017), as well as Amdeberhan, Andrews and Ballantine (J Combin Theory Series A 221:106154, 2026). The proofs are based on classical transformations in the theory of infinite series together with a systematic rearrangement of double Lambert series.