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<p>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 <i>q</i>-series by Ramanujan. Our main emphasis will be on explicit representations for a variety of <i>q</i>-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.</p>

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Recursive formulas for MacMahon and Ramanujan q-series

  • 期刊论文

  • 发表日期：

    2025年4月3日

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  • Tewodros Amdeberhan,
  • Rupam Barman,
  • Ajit Singh

摘要

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.