Ramanujan--Fine integrals for level 10
Shaun Cooper, Timothy Huber, Jeffery Opoku
TL;DR
The paper addresses when eta quotients arise as derivatives of integer-coefficient power series, focusing on level 10 to obtain Ramanujan--Fine integrals. It combines a computer-augmented search for eta-quotient derivatives with analytic tools anchored in Ramanujan’s level-10 parameter $k$, hypergeometric limits, and Ramanujan–Fine-type identities to produce explicit evaluations, including an infinite family. Eight level-10 instances and three infinite families are identified, with a proof-of-concept showing integrality via $k$-parametrizations and a rigorous limit analysis; an explicit value for $k(e^{-2\pi/\sqrt{10}})$ is given in the appendix. The work advances a constructive method for level-10 Ramanujan--Fine integrals, offers conjectures toward a complete classification, and enriches the interplay between eta-quotients, modular-like structures, and integer-coefficient series.
Abstract
We investigate the question of when an eta quotient is a derivative of a formal power series with integer coefficients and present an analysis in the case of level 10. As a consequence, we establish and classify an infinite number of integral evaluations such as $$ \int_0^{e^{-2π/\sqrt{10}}} q\prod_{j=1}^\infty \frac{(1-q^j)^3(1-q^{10j})^8}{(1-q^{5j})^7} \text{d} q = \frac14\left(\sqrt{10-4\sqrt{5}}-1\right). $$ We describe how the results were found and give reasons for why it is reasonable to conjecture that the list is complete for level 10.