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

DOI:
https://doi.org/10.1090/S0002-9939-09-09896-7

Published electronically:
May 6, 2009

MathSciNet review:
2506454

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Abstract | References | Similar Articles | Additional Information

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

**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**, https://doi.org/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**, https://doi.org/10.1017/S0017089500006194

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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

DOI:
https://doi.org/10.1090/S0002-9939-09-09896-7

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.