Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Lacunarity of certain partition-theoretic generating functions

Authors: Emily Clader, Yvonne Kemper and Matt Wage
Journal: Proc. Amer. Math. Soc. 137 (2009), 2959-2968
MSC (2000): Primary 11F30, 11P82, 11F11; Secondary 11F20
Published electronically: May 6, 2009
MathSciNet review: 2506454
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a certain family of infinite products, denoted $ f_{a,b}$, which were introduced by Han as a generalization of the Nekrasov-Okounkov formula. Extending the work of Serre on powers of Dedekind's $ \eta$-function, we investigate the integers $ a$ and $ b$ for which ``almost all'' of the Fourier coefficients of $ f_{a,b}$ are zero (forms with this property are referred to as lacunary). We give the complete list of pairs $ (a,b)$, where $ b$ is odd, for which $ f_{a,b}$ is lacunary.

References [Enhancements On Off] (What's this?)

  • 1. Tom M. Apostol, Modular functions and Dirichlet series in number theory, 2nd ed., Graduate Texts in Mathematics, vol. 41, Springer-Verlag, New York, 1990. MR 1027834
  • 2. Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids 1, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507–530 (1975) (French). MR 0379379
  • 3. Frank Garvan, Dongsu Kim, and Dennis Stanton, Cranks and 𝑡-cores, Invent. Math. 101 (1990), no. 1, 1–17. MR 1055707, 10.1007/BF01231493
  • 4. G-N. Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension, and applications (preprint).
  • 5. Henryk Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964
  • 6. Neal Koblitz, Introduction to elliptic curves and modular forms, 2nd ed., Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1993. MR 1216136
  • 7. Ken Ono, Gordon’s 𝜀-conjecture on the lacunarity of modular forms, C. R. Math. Acad. Sci. Soc. R. Can. 20 (1998), no. 4, 103–107 (English, with English and French summaries). MR 1662100
  • 8. Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and 𝑞-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489
  • 9. J-P. Serre, Quelques applications du théorème de densité de Chebatorev, Publ. Math. I.H.E.S. 54 (1981), pp. 123-201.
  • 10. Jean-Pierre Serre, Sur la lacunarité des puissances de 𝜂, Glasgow Math. J. 27 (1985), 203–221 (French). MR 819840, 10.1017/S0017089500006194

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11F30, 11P82, 11F11, 11F20

Retrieve articles in all journals with MSC (2000): 11F30, 11P82, 11F11, 11F20

Additional Information

Emily Clader
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027

Yvonne Kemper
Affiliation: Department of Mathematics, University of California Berkeley, Berkeley, California 94708

Matt Wage
Affiliation: Appleton East High School, 1411 N. Briarcliff Drive, Appleton, Wisconsin 54915
Address at time of publication: Princeton University, 0920 Frist Campus Center, Princeton, New Jersey 08544

Received by editor(s): July 31, 2008
Received by editor(s) in revised form: January 19, 2009
Published electronically: May 6, 2009
Communicated by: Jim Haglund
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.