Congruences of multipartition functions modulo powers of primes |
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Authors: | William Y C Chen Daniel K Du Qing-Hu Hou Lisa H Sun |
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Institution: | 1. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, P.R. China 2. Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin, 300071, P.R. China
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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) γ ?(24z), where η(z) is the Dedekind’s eta function and ?(z) is a holomorphic modular form in \(M_{\lambda}(\mathrm{SL}_{2}(\mathbb{Z}))\) . 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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