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A REMARK ON GENERALISED CUBIC PARTITIONS MODULO
$5$
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 07 May 2025, pp. 1-5
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Amdeberhan et al. [‘Arithmetic properties for generalized cubic partitions and overpartitions modulo a prime’, Aequationes Math. (2024), doi:10.1007/s00010-024-01116-7] defined the generalised cubic partition function
$a_c(n)$ as the number of partitions of n whose even parts may appear in 
$c\geq 1$ different colours and proved that 
$a_3(7n+4)\equiv 0\pmod {7}$ and 
$a_5(11n+10)\equiv 0\pmod {11}$ for all 
$n\geq 0$ via modular forms. Recently, the author [‘A note on congruences for generalized cubic partitions modulo primes’, Integers 25 (2025), Article no. A20] gave elementary proofs of these congruences. We prove in this note two infinite families of congruences modulo 
$5$ for 
$a_c(n)$ given by 
$$ \begin{align*} a_{25c+4}(25n+18)&\equiv 0\pmod{5},\\ a_{25c+9}(25n+8)&\equiv 0\pmod{5} \end{align*} $$for all
$c\geq 0$ and 
$n\geq 0$ using elementary q-series manipulations.
VANISHING COEFFICIENTS IN QUOTIENTS OF THETA FUNCTIONS OF MODULUS FIVE
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 102 / Issue 3 / December 2020
- Published online by Cambridge University Press:
- 27 March 2020, pp. 387-398
- Print publication:
- December 2020
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