RANK AND CONGRUENCES FOR OVERPARTITION PAIRS

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RANK DIFFERENCES FOR OVERPARTITIONS (2007, background): “The rank of a partition studied by Atkin and Swinnerton-Dyer is now understood to be a special case of a more general rank which is dened on overpartition pairs [ 6 ].”
DYSON'S RANK, OVERPARTITIONS, AND WEAK MAASS FORMS (2007, background): “It is worth mentioning that for the case of the classical Dyson’s rank generating functions in [9], the weak Maass forms have weight 1/2 for every root of unity 6= 1. 3) The authors [ 7 ] show that in the context of overpartition pairs, the analogous generating functions associated to the appropriat”
The spt-crank for overpartitions (2013, methodology): “The generating functions for the spt functions are given as follows; these are special cases of a general SPT function due to Bringmann, Lovejoy, and Osburn [15, Section 7], SPT( d, e ; q ) = ( − dq ; q ) ∞ ( − eq ; q ) ∞ ( deq ; q ) ∞ ( q ; q ) ∞ ∞ n =1 q n ( q ; q ) n ( deq ; q ) n (1 − q n ) 2 ( ”