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arXiv · 2412.00262

Recursive Formulas for MacMahon and Ramanujan $q$-series

Tewodros Amdeberhan·Rupam Barman·Ajit Singh
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Abstract

In the present work, we extend current research in a nearly-forgotten but newly revived topic, initiated by P. A. MacMahon, on a generalized notion which relates the divisor sums to the theory of integer partitions and two infinite families of $q$-series by Ramanujan. Our main emphasis will be on explicit representations for a variety of $q$-series, studied primarily by MacMahon and Ramanujan, with an eye towards their modular properties and their proper place in the ring of quasimodular forms of level one and level two.

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Tewodros Amdeberhan, Rupam Barman, Ajit Singh. 2024-11-29. Recursive Formulas for MacMahon and Ramanujan $q$-series. https://arxiv.org/abs/2412.00262

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