Congruences of Multipartition Functions Modulo Powers of Primes
William Y.C. Chen, Daniel K. Du, Qing-Hu Hou and Lisa H.Sun
Abstract: Let pr(n) denote the number of r-component multipartitions of n, and let Sγ λ be the space spanned by η(24z)γφ(24z), where η(z) is the Dedekind’s eta function and η(z) is a holomorphic modular form in Mλ(SL2(Z)). In this paper, we show that the generating function of pr with respect to n is congruent to a function in the space Sγ, λ modulo mk. 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 p2(n) modulo 5 and p8(n) modulo 11. Furthermore, using the invariance property of Sγ, λ under the Hecke operator , we obtain two classes of congruences pertaining to the mk-adic property of pr(n).
AMS Classification: 05A17, 11F33, 11P83.
Keywords: modular form, partition, multipartition, Ramanujan-type congruence