A Resolvent Approach to Generalized Lambert Series and q-Series Identities

2025/07/20 by Ronald Orozco López, López, Ronald Orozco
Engineering · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Experimental and Theoretical Physics Studies #Sports Dynamics and Biomechanics #math.NT

paper · pdf · doi:10.48550/arxiv.2507.15160

openalex publication_date 2025/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a partial-theta-type \(q\)-operator Θ(yD_q)=∑_n≥0q^{\binom n2}yⁿD_qⁿ, D_qf(x)=(f(qx))/(x), and show that it admits the resolvent representation Θ(yD_q)=(I-\frac yxM_q)^-1, where M_qf(x)=f(qx). This identity provides a unified operational framework for generalized Lambert series and their Mehler, Rogers, and bilateral analogues. Starting from ordinary and bilateral generating functions, we obtain Lambert-type expansions and derive consequences involving basic hypergeometric series, Ramanujan's ₁ψ₁ summation, and Kronecker-type theta identities. The method gives a compact way to generate families of q-series identities from a first-order q-difference resolvent.

Citations

• Chapter 16 of Ramanujan’s second notebook: theta-functions and 𝑞-series

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