M 2 -rank differences for partitions without repeated odd parts

Jeremy Lovejoy[1]; Robert Osburn[2]

  • [1] CNRS LIAFA Université Denis Diderot 2, Place Jussieu, Case 7014 F-75251 Paris Cedex 05, FRANCE
  • [2] School of Mathematical Sciences University College Dublin Belfield Dublin 4

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 2, page 313-334
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Lovejoy, Jeremy, and Osburn, Robert. "$M_2$-rank differences for partitions without repeated odd parts." Journal de Théorie des Nombres de Bordeaux 21.2 (2009): 313-334. <http://eudml.org/doc/10883>.

@article{Lovejoy2009,
abstract = {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.},
affiliation = {CNRS LIAFA Université Denis Diderot 2, Place Jussieu, Case 7014 F-75251 Paris Cedex 05, FRANCE; School of Mathematical Sciences University College Dublin Belfield Dublin 4},
author = {Lovejoy, Jeremy, Osburn, Robert},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {partitions; modular forms; generalized Lambert series},
language = {eng},
number = {2},
pages = {313-334},
publisher = {Université Bordeaux 1},
title = {$M_2$-rank differences for partitions without repeated odd parts},
url = {http://eudml.org/doc/10883},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Lovejoy, Jeremy
AU - Osburn, Robert
TI - $M_2$-rank differences for partitions without repeated odd parts
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 2
SP - 313
EP - 334
AB - 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.
LA - eng
KW - partitions; modular forms; generalized Lambert series
UR - http://eudml.org/doc/10883
ER -

References

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  1. G. E. Andrews, A generalization of the Göllnitz-Gordon partition theorems. Proc. Amer. Math. Soc. 8 (1967), 945–952. Zbl0153.32801MR219497
  2. G.E. Andrews, Two theorems of Gauss and allied identities proven arithmetically. Pacific J. Math. 41 (1972), 563–578. Zbl0219.10021MR349572
  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. Zbl0482.33002MR654518
  4. G.E. Andrews, Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks. Invent. Math. 169 (2007), 37–73. Zbl1214.11116MR2308850
  5. G.E. Andrews and B.C. Berndt, Ramanujan’s Lost Notebook, Part I. Springer, New York, 2005. Zbl1075.11001
  6. A.O.L. Atkin and H. Swinnerton-Dyer, Some properties of partitions. Proc. London Math. Soc. 66 (1954), 84–106. Zbl0055.03805MR60535
  7. A.O.L. Atkin and S.M. Hussain, Some properties of partitions. II. Trans. Amer. Math. Soc. 89 (1958), 184–200. Zbl0083.26201MR103872
  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. Zbl1016.05003MR1932070
  9. K. Bringmann, K. Ono, and R. Rhoades, Eulerian series as modular forms. J. Amer. Math. Soc. 21 (2008), 1085–1104. Zbl1208.11065MR2425181
  10. S.H. Chan, Generalized Lambert series identities. Proc. London Math. Soc. 91 (2005), 598–622. Zbl1089.33012MR2180457
  11. S. Corteel and O. Mallet, Overpartitions, lattice paths, and Rogers-Ramanujan identities. J. Combin. Theory Ser. A 114 (2007), 1407–1437. Zbl1126.11056MR2360678
  12. F.G. Garvan, Generalizations of Dyson’s rank and non-Rogers-Ramanujan partitions. Manuscripta Math. 84 (1994), 343–359. Zbl0819.11043MR1291125
  13. G. Gasper and M. Rahman, Basic Hypergeometric Series. Cambridge Univ. Press, Cambridge, 1990. Zbl0695.33001MR1052153
  14. B. Gordon and R.J. McIntosh, Some eighth order mock theta functions. J. London Math. Soc. 62 (2000), 321–335. Zbl1031.11007MR1783627
  15. D. Hickerson, A proof of the mock theta conjectures. Invent. Math. 94 (1988), 639–660. Zbl0661.10059MR969247
  16. J. Lovejoy, Rank and conjugation for a second Frobenius representation of an overpartition. Ann. Comb. 12 (2008), 101–113. Zbl1147.11062MR2401139
  17. J. Lovejoy and R. Osburn, Rank differences for overpartitions. Quart. J. Math. (Oxford) 59 (2008), 257–273. Zbl1153.11052MR2428080
  18. P.A. MacMahon, The theory of modular partitions. Proc. Cambridge Phil. Soc. 21 (1923), 197–204. Zbl48.0154.01
  19. R.J. McIntosh, Second order mock theta functions. Canad. Math. Bull. 50 (2) (2007), 284–290. Zbl1133.11013MR2317449

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