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A new analogue of $t$-core partitions

Volume 199 / 2021

D. S. Gireesh, Chiranjit Ray, C. Shivashankar Acta Arithmetica 199 (2021), 33-53 MSC: 05A15, 05A17, 11F11, 11F20, 11F33. DOI: 10.4064/aa200516-10-12 Published online: 4 March 2021

Abstract

By analogy with $t$-core partitions, we study $\overline {a}_t(n)$, given by $\sum _{n=0}^{\infty }\overline {a}_t(n)q^n\break ={\phi (-q^t)^t}/{\phi (-q)}.$ We obtain multiplicative formulas and arithmetic properties of $\overline {a}_{t}(n)$ for $t\in \{3,4,8\}$. Moreover, if $8n+5$ is square-free then we prove $\overline {a}_{4}(2^{2\alpha }n)=12h(-4n)$, where $\alpha $ is any positive integer and $h(D)$ denotes the class number of binary quadratic forms of discriminant $D$. For a fixed positive integer $j$ and prime numbers $p_i\geq 5,$ we also show that $\overline {a}_{t}(n)$ is almost always divisible by $p_i^j$ if $p_i^{2a_i}\geq t$, where $t=p_1^{a_1} \dots p_m^{a_m}.$ Additionally, using modular forms we prove a Ramanujan type congruence for $\overline {a}_{5}$ modulo $5$.

Authors

  • D. S. GireeshDepartment of Mathematics and Statistics
    M. S. Ramaiah University of Applied Sciences
    Peenya 4th phase, Peenya
    Bengaluru 560 058, Karnataka, India
    e-mail
  • Chiranjit RayDepartment of Mathematics
    Harish-Chandra Research Institute
    Prayagraj 211 019, Uttar Pradesh, India
    e-mail
  • C. ShivashankarDepartment of Mathematics and Statistics
    M. S. Ramaiah University of Applied Sciences
    Peenya 4th phase, Peenya
    Bengaluru 560 058, Karnataka, India
    e-mail

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