Congruences of Multipartition Functions Modulo Powers of Primes
William Y.C. Chen, Daniel K. Du, Qing-Hu Hou and Lisa H.Sun
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}φ(24z), where η(z) is the Dedekind’s eta function and η(z) is a holomorphic modular form in M_{λ}(SL_{2}(Z)). In this paper, we show that the generating function of p_{r} 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 , we obtain two classes of congruences pertaining to the m^{k}-adic property of p_{r}(n). AMS Classification: 05A17, 11F33, 11P83. Keywords: modular form, partition, multipartition, Ramanujan-type congruence Download: pdf |