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Congruences for modular forms and applications to crank functions

Volume 215 / 2024

Hao Zhang, Helen W. J. Zhang Acta Arithmetica 215 (2024), 73-83 MSC: Primary 05A17; Secondary 11P83 DOI: 10.4064/aa231026-24-1 Published online: 29 April 2024

Abstract

Motivated by the work of Mahlburg, which refined the work of Ono, we find congruences for a large class of modular forms. Moreover, we generalize the generating function of the Andrews–Garvan–Dyson crank of partitions and establish several new infinite families of congruences. In this framework, we show that both the birank of an ordered pair of partitions introduced by Hammond and Lewis, and $k$-crank of $k$-colored partitions introduced by Fu and Tang, have the same properties as the partition function and crank.

Authors

  • Hao ZhangSchool of Mathematics
    Hunan University
    Changsha 410082, P.R. China
    and
    Hunan Provincial Key Laboratory of
    Intelligent Information Processing and Applied Mathematics
    Changsha 410082, P.R. China
    e-mail
  • Helen W. J. ZhangSchool of Mathematics
    Hunan University
    Changsha 410082, P.R. China
    and
    Hunan Provincial Key Laboratory of
    Intelligent Information Processing and Applied Mathematics
    Changsha 410082, P.R. China
    e-mail

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