On the rank and the crank modulo $4$ and $8$

Abstract

In this paper we prove some identities, conjectured by Lewis, for the rank and crank of partitions concerning the modulo 4 and 8. These identities are similar to Dyson's identities for the rank modulo 5 and 7 which give a combinatorial interpretation to Ramanujan's partition congruences. For this, we use multisection of series and some of the results that Watson established for the third order mock theta functions.