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arXiv · 1206.6642

Congruences of Multipartition Functions Modulo Powers of Primes

William Y. C. Chen·Daniel K. Du·Qing-Hu Hou·Lisa H. Sun
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Abstract

Let $p_r(n)$ denote the number of $r$-component multipartitions of $n$, and let $S_{γ,λ}$ be the space spanned by $η(24z)^γϕ(24z)$, where $η(z)$ is the Dedekind's eta function and $ϕ(z)$ is a holomorphic modular form in $M_λ({\rm SL}_2(\mathbb{Z}))$. In this paper, we show that the generating function of $p_r(\frac{m^k n +r}{24})$ with respect to $n$ is congruent to a function in the space $S_{γ,λ}$ modulo $m^k$. As special cases, this relation leads to many well known congruences including the Ramanujan congruences of $p(n)$ modulo $5,7,11$ and Gandhi's congruences of $p_2(n)$ modulo 5 and $p_{8}(n)$ modulo 11. Furthermore, using the invariance property of $S_{γ,λ}$ under the Hecke operator $T_{\ell^2}$, we obtain two classes of congruences pertaining to the $m^k$-adic property of $p_r(n)$.

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William Y. C. Chen, Daniel K. Du, Qing-Hu Hou, Lisa H. Sun. 2012-06-28. Congruences of Multipartition Functions Modulo Powers of Primes. https://arxiv.org/abs/1206.6642

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