Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T07:30:05.991Z Has data issue: false hasContentIssue false
  • English
  • Français

Automorphic properties of generating functions for generalized odd rank moments and odd Durfee symbols

Published online by Cambridge University Press:  18 July 2011

CLAUDIA ALFES ,
KATHRIN BRINGMANN  and
JEREMY LOVEJOY
CLAUDIA ALFES
Affiliation:
Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany. e-mail: [email protected]
KATHRIN BRINGMANN
Affiliation:
Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany. e-mail: [email protected]
JEREMY LOVEJOY
Affiliation:
CNRS, LIAFA, Université Denis Diderot, Case 7014, 75205 Paris Cedex 13, France. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We define two-parameter generalizations of Andrews' (k+1)-marked odd Durfee symbols and 2kth symmetrized odd rank moments, and study the automorphic properties of some of their generating functions. When k = 0 we obtain families of modular forms and mock modular forms. When k ≥ 1, we find quasimodular forms and quasimock modular forms.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 151 , Issue 3 , November 2011 , pp. 385 - 406
Copyright
Copyright © Cambridge Philosophical Society 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Andrews, G. Mordell integrals and Ramanujan's “lost” notebook. Lecture Notes in Math. 899 (Springer, 1981), 10–48.Google Scholar
[2]Andrews, G.Bailey chains and generalized Lambert series: I. Four identities of Ramanujan. Illinois J. Math. 36 (1992), 251–274.Google Scholar
[3]Andrews, G.Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks. Invent. Math. 169 (2007), 37–73.CrossRefGoogle Scholar
[4]Bringmann, K. and Lovejoy, J.Overpartitions and class numbers of binary quadratic forms. Proc. Nat. Acad. Sci. USA 106 (2009), 5513–5516.CrossRefGoogle ScholarPubMed
[5]Bringmann, K., Lovejoy, J. and Osburn, R. Automorphic properties of generating functions for generalized rank moments and Durfee symbols. Int. Math. Res. Not. (2010), rnp131.CrossRefGoogle Scholar
[6]Bringmann, K., Lovejoy, J. and Osburn, R.Rank and crank moments for overpartitions. J. Number Theory 129 (2009), 2567–2574.CrossRefGoogle Scholar
[7]Bringmann, K., Ono, K. and Rhoades, R.Eulerian series as modular forms, J. Amer. Math. Soc. 21 (2008), 1085–1104.Google Scholar
[8]Bringmann, K. and Richter, O.Zagier-type dualities and lifting maps for harmonic Maass-Jacobi forms. Adv. Math. 225 (2010), 2298–2315.Google Scholar
[9]Bringmann, K. and Zwegers, S.Rank-crank type PDE's and non-holomorphic Jacobi forms. Math. Res. Lett. 17 (2010), 589–600.Google Scholar
[10]Bruinier, J. and Funke, J.On two geometric theta lifts. Duke Math. J. 125 (2004), 45–90.Google Scholar
[11]Cohen, H.A course in computational algebraic number theory. Graduate Texts in Math. 138 (Springer-Verlag, 1993).CrossRefGoogle Scholar
[12]Eichler, M. and Zagier, D.Jacobi Forms. Progr. in Math. 55 (Birkhäuser, 1985).Google Scholar
[13]Gasper, G. and Rahman, M.Basic hypergeometric series (Cambridge University Press, 1990).Google Scholar
[14]Golubeva, E. P. and Fomenko, O. M.Series ∑F(m)q m, where F(m) is the number of odd classes of binary quadratic forms of determinant −m. J. Math. Sci. 17 (1981), 1759–1766.CrossRefGoogle Scholar
[15]Gordon, B. and McIntosh, R.Some eighth order mock theta functions. J. London Math. Soc. 62 (2000), 321–335.Google Scholar
[16]Hikami, K.Transformation of the “second” order mock theta function. Lett. Math. Phys. 75 (2006), 93–98.Google Scholar
[17]Humbert, G.Formules relatives aux nombres de classes des formes quadratiques binaires et positives. J. Math. Pures Appl. (6) 3 (1907), 337–449.Google Scholar
[18]Kronecker, M.Über die Anzahl der verschiedenen Classen quadratischer Formen von negativer Determinante. J. Reine Angew. Math. 57 (1860), 248–255.Google Scholar
[19]McIntosh, R.Second order mock theta functions. Canad. Math. Bull. 50 (2007), 284–290.CrossRefGoogle Scholar
[20]Ono, K. The web of modularity: arithmetic of the coefficients of modular forms and q-series. CBMS Regional Conference Series in Mathematics, 102 (Amer. Math. Soc., Providence, RI, 2004).Google Scholar
[21]Ono, K. Unearthing the visions of a master: harmonic Maass forms and number theory. 2008 Harvard-MIT Current Developments in Mathematics Conference (Int. Press, Somerville, MA, 2009), 347–454.Google Scholar
[22]Ramanujan, S.The Lost Notebook and Other Unpublished Papers, (Narosa Publishing House, New Delhi, 1988).Google Scholar
[23]Watson, G.The final problem: An account of the mock theta functions. J. London Math. Soc. 11 (1936), 55–80.Google Scholar
[24]Watson, G.Generating functions of class-numbers. Compositio Math. 1 (1935), 39–68.Google Scholar
[25]Zagier, D. The dilogarithm function. Frontiers in Number Theory, Physics and Geometry, II. (Springer, 2007) 3–65.Google Scholar
[26]Zagier, D. Ramanujan's mock theta functions and their applications d'après Zwegers and Bringmann-Ono. Séminaire Bourbaki 60ème Année (986) (2006–2007), http://www.bourbaki.ens.fr/TEXTES/986.pdf.Google Scholar
[27]Zwegers, S.Mock ϑ-functions and real analytic modular forms, q-series with applications to combinatorics, number theory, and physics (Ed. B. C. Berndt and K. Ono). Contemp. Math. 291 (Amer. Math. Soc., 2001), 269–277.Google Scholar
[28]Zwegers, S. Mock Theta Functions, PhD thesis, Utrecht, 2002.Google Scholar