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Eta-quotients and divisibility of certain partition functions by powers of primes

Authors:
Ajit Singh at University of Virginia
Ajit Singh
Rupam Barman at Indian Institute of Technology Guwahati
Rupam Barman
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Abstract

Andrews' (k,i)(k, i)-singular overpartition function C‾k,i(n)\overline{C}_{k, i}(n) counts the number of overpartitions of n in which no part is divisible by k and only parts ≡±i(modk)\equiv \pm i\pmod{k} may be overlined. In recent times, divisibility of C‾3ℓ,ℓ(n)\overline{C}_{3\ell, \ell}(n), C‾4ℓ,ℓ(n)\overline{C}_{4\ell, \ell}(n) and C‾6ℓ,ℓ(n)\overline{C}_{6\ell, \ell}(n) by 2 and 3 are studied for certain values of ℓ\ell. In this article, we study divisibility of C‾3ℓ,ℓ(n)\overline{C}_{3\ell, \ell}(n), C‾4ℓ,ℓ(n)\overline{C}_{4\ell, \ell}(n) and C‾6ℓ,ℓ(n)\overline{C}_{6\ell, \ell}(n) by primes p≥5p\geq 5. For all positive integer ℓ\ell and prime divisors p≥5p\geq 5 of ℓ\ell, we prove that C‾3ℓ,ℓ(n)\overline{C}_{3\ell, \ell}(n), C‾4ℓ,ℓ(n)\overline{C}_{4\ell, \ell}(n) and C‾6ℓ,ℓ(n)\overline{C}_{6\ell, \ell}(n) are almost always divisible by arbitrary powers of p. For s∈{3,4,6}s\in \{3, 4, 6\}, we next show that the set of those n for which C‾s⋅ℓ,ℓ(n)≢0(modpik)\overline{C}_{s\cdot\ell, \ell}(n) \not\equiv 0\pmod{p_i^k} is infinite, where k is a positive integer satisfying pik−1≥ℓp_i^{k-1}\geq \ell. We further improve a result of Gordon and Ono on divisibility of ℓ\ell-regular partitions by powers of certain primes. We also improve a result of Ray and Chakraborty on divisibility of ℓ\ell-regular overpartitions by powers of certain primes.

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