M 2 -rank differences for partitions without repeated odd parts
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 2, p. 313-334

We prove formulas for the generating functions for M 2 -rank differences for partitions without repeated odd parts. These formulas are in terms of modular forms and generalized Lambert series.

Nous prouvons des formules pour les fonctions génératrices des différences de rang M 2 pour les partitions où les parts impaires sont distinctes. Ces formules sont en termes de formes modulaires et de séries de Lambert généralisées.

@article{JTNB_2009__21_2_313_0,
     author = {Lovejoy, Jeremy and Osburn, Robert},
     title = {$M\_2$-rank differences for partitions without repeated odd parts},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {2},
     year = {2009},
     pages = {313-334},
     doi = {10.5802/jtnb.673},
     mrnumber = {2541428},
     zbl = {pre05620653},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2009__21_2_313_0}
}
Lovejoy, Jeremy; Osburn, Robert. $M_2$-rank differences for partitions without repeated odd parts. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 2, pp. 313-334. doi : 10.5802/jtnb.673. http://www.numdam.org/item/JTNB_2009__21_2_313_0/

[1] G. E. Andrews, A generalization of the Göllnitz-Gordon partition theorems. Proc. Amer. Math. Soc. 8 (1967), 945–952. | MR 219497 | Zbl 0153.32801

[2] G.E. Andrews, Two theorems of Gauss and allied identities proven arithmetically. Pacific J. Math. 41 (1972), 563–578. | MR 349572 | Zbl 0219.10021

[3] G.E. Andrews, The Mordell integrals and Ramanujan’s “lost” notebook. In: Analytic Numbere Theory, Lecture Notes in Mathematics 899. Springer-Verlag, New York, 1981, pp. 10–48. | MR 654518 | Zbl 0482.33002

[4] G.E. Andrews, Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks. Invent. Math. 169 (2007), 37–73. | MR 2308850 | Zbl pre05353177

[5] G.E. Andrews and B.C. Berndt, Ramanujan’s Lost Notebook, Part I. Springer, New York, 2005. | Zbl 1075.11001

[6] A.O.L. Atkin and H. Swinnerton-Dyer, Some properties of partitions. Proc. London Math. Soc. 66 (1954), 84–106. | MR 60535 | Zbl 0055.03805

[7] A.O.L. Atkin and S.M. Hussain, Some properties of partitions. II. Trans. Amer. Math. Soc. 89 (1958), 184–200. | MR 103872 | Zbl 0083.26201

[8] A. Berkovich and F. G. Garvan, Some observations on Dyson’s new symmetries of partitions. J. Combin. Theory, Ser. A 100 (2002), 61–93. | MR 1932070 | Zbl 1016.05003

[9] K. Bringmann, K. Ono, and R. Rhoades, Eulerian series as modular forms. J. Amer. Math. Soc. 21 (2008), 1085–1104. | MR 2425181

[10] S.H. Chan, Generalized Lambert series identities. Proc. London Math. Soc. 91 (2005), 598–622. | MR 2180457 | Zbl 1089.33012

[11] S. Corteel and O. Mallet, Overpartitions, lattice paths, and Rogers-Ramanujan identities. J. Combin. Theory Ser. A 114 (2007), 1407–1437. | MR 2360678 | Zbl 1126.11056

[12] F.G. Garvan, Generalizations of Dyson’s rank and non-Rogers-Ramanujan partitions. Manuscripta Math. 84 (1994), 343–359. | MR 1291125 | Zbl 0819.11043

[13] G. Gasper and M. Rahman, Basic Hypergeometric Series. Cambridge Univ. Press, Cambridge, 1990. | MR 1052153 | Zbl 0695.33001

[14] B. Gordon and R.J. McIntosh, Some eighth order mock theta functions. J. London Math. Soc. 62 (2000), 321–335. | MR 1783627 | Zbl 1031.11007

[15] D. Hickerson, A proof of the mock theta conjectures. Invent. Math. 94 (1988), 639–660. | MR 969247 | Zbl 0661.10059

[16] J. Lovejoy, Rank and conjugation for a second Frobenius representation of an overpartition. Ann. Comb. 12 (2008), 101–113. | MR 2401139 | Zbl 1147.11062

[17] J. Lovejoy and R. Osburn, Rank differences for overpartitions. Quart. J. Math. (Oxford) 59 (2008), 257–273. | MR 2428080 | Zbl 1153.11052

[18] P.A. MacMahon, The theory of modular partitions. Proc. Cambridge Phil. Soc. 21 (1923), 197–204.

[19] R.J. McIntosh, Second order mock theta functions. Canad. Math. Bull. 50 (2) (2007), 284–290. | MR 2317449 | Zbl 1133.11013