Derivatives of theta functions as Traces of Partition Eisenstein series

2024/07/11 by Tewodros Amdeberhan, Ken Ono, Amdeberhan, Tewodros +3 · 4 citations
Mathematics · #05A17 #11F03 #11M36 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2407.08437

openalex publication_date 2024/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In his "lost notebook'', Ramanujan used iterated derivatives of two theta functions to define sequences of q-series \U_2t(q)\ and \V_2t(q)\ that he claimed to be quasimodular. We give the first explicit proof of this claim by expressing them in terms of "partition Eisenstein series'', extensions of the classical Eisenstein series E_2k(q) defined by λ=(1^m₁, 2^m₂,…, n^m_n) \vdash n \longmapsto E_λ(q):= E₂(q)^m₁ E₄(q)^m₂⋯ E_2n(q)^m_n. For functions ϕ: P↦ ℂ on partitions, the weight 2n partition Eisenstein trace is Tr_n(ϕ;q):=∑_{λ\vdash n} ϕ(λ)E_λ(q). For all t, we prove that U_2t(q)=Tr_t(ϕ_U;q) and V_2t(q)=Tr_t(ϕ_V;q), where ϕ_U and ϕ_V are natural partition weights, giving the first explicit quasimodular formulas for these series.

Cited by

• Mock modular forms from the k-rank moments
• Multiple \wp-Functions and Their Applications
• Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials
• Applications of Faà di Bruno's formula to partition traces

Related

• Traces of partition Eisenstein series
• Traces of partition Eisenstein series and almost holomorphic modular forms
• Applications of Ramanujan's work on Eisenstein series
• Eichler integrals of Eisenstein series as q-brackets of weighted t-hook functions on partitions
• The Eichler integral of E₂ and q-brackets of t-hook functions