Article ID Journal Published Year Pages File Type
4593997 Journal of Number Theory 2013 10 Pages PDF
Abstract
Andrews, Lewis and Lovejoy introduced the partition function PD(n) as the number of partitions of n with designated summands, where we assume that among parts with equal size, exactly one is designated. They proved that PD(3n+2) is divisible by 3 and showed that the generating function of PD(3n) can be expressed as an infinite product of powers of (1−q2n+1) times a function F(q2). We obtain a Ramanujan type identity which implies the congruence for PD(3n+2). We also find an explicit formula for F(q2), which leads to a formula for the generating function of PD(3n). A formula for the generating function of PD(3n+1) is also obtained. Our proofs rely on Chanʼs identity on Ramanujanʼs cubic continued fraction and identities on cubic theta functions. By introducing a rank for partitions with designated summands, we give a combinatorial interpretation of the congruence for PD(3n+2).
Keywords
Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory
Authors
, , , ,