Rank differences for overpartitions

Abstract

In 1954, Atkin and Swinnerton-Dyer proved Dyson's conjectures on the rank of a partition by establishing formulas for the generating functions for rank differences in arithmetic progressions. In this paper, we prove formulas for the generating functions for rank differences for overpartitions. These are in terms of modular functions and generalized Lambert series.

17 pages, final version, accepted for publication in the Quarterly Journal of Mathematics

References in corpus (4)

• Mock Theta Functions
• Rank and crank moments for overpartitions
• M_2-rank differences for partitions without repeated odd parts
• Overpartitions, lattice paths and Rogers-Ramanujan identities

Cited by in corpus (11)

• M_2-rank differences for partitions without repeated odd parts
• Dyson's Ranks and Appell-Lerch Sums
• Automorphic properties of generating functions for generalized rank moments and Durfee symbols
• Exotic Bailey-Slater SPT-Functions II: Hecke-Rogers-Type Double Sums and Bailey Pairs From Groups A, C, E
• Ranks of overpartitions: Asymptotics and inequalities
• The Unimodality of the Crank on Overpartitions
• Efficient computation of the overpartition function and applications
• On recursions for coefficients of mock theta functions
• Monotonicity properties for ranks of overpartitions
• Rank deviations for overpartitions
• Combinatorial Interpretations of Cranks of Overpartitions and Partitions without Repeated Odd Parts