Congruences of Multipartition Functions Modulo Powers of Primes

2012/06/28 by William Y. C. Chen, Daniel K. Du, Chen, William Y. C. +5
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Mathematical functions and polynomials #math.CO #math.NT #msc:05A17 #msc:11F33 #msc:11P83

paper · pdf · doi:10.48550/arxiv.1206.6642

19 pages

arxiv created 2012/06/28 · arxiv updated 2012/06/29

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 SL₂(ℤ)). In this paper, we show that the generating function of p_r((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 p₂(n) modulo 5 and p₈(n) modulo 11. Furthermore, using the invariance property of S_γ,λ under the Hecke operator T_ℓ², we obtain two classes of congruences pertaining to the m^k-adic property of p_r(n).

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