## Eulerian series as modular forms

HTML articles powered by AMS MathViewer

- by Kathrin Bringmann, Ken Ono and Robert C. Rhoades;
- J. Amer. Math. Soc.
**21**(2008), 1085-1104 - DOI: https://doi.org/10.1090/S0894-0347-07-00587-5
- Published electronically: November 26, 2007
- PDF | Request permission

## Abstract:

In 1988, Hickerson proved the celebrated â€śmock theta conjecturesâ€ť in a collection of ten identities from Ramanujanâ€™s â€ślost notebookâ€ť which express certain modular forms as linear combinations of mock theta functions. In the context of Maass forms, these identities arise from the peculiar phenomenon that two different harmonic Maass forms may have the same non-holomorphic parts. Using this perspective, we construct several infinite families of modular forms which are differences of mock theta functions.## References

- George E. Andrews,
*$q$-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra*, CBMS Regional Conference Series in Mathematics, vol. 66, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR**858826**, DOI 10.1090/cbms/066 - George E. Andrews,
*Mock theta functions*, Theta functionsâ€”Bowdoin 1987, Part 2 (Brunswick, ME, 1987) Proc. Sympos. Pure Math., vol. 49, Amer. Math. Soc., Providence, RI, 1989, pp.Â 283â€“298. MR**1013178**, DOI 10.1090/pspum/049.2/1013178 - George E. Andrews and F. G. Garvan,
*Ramanujanâ€™s â€ślostâ€ť notebook. VI. The mock theta conjectures*, Adv. in Math.**73**(1989), no.Â 2, 242â€“255. MR**987276**, DOI 10.1016/0001-8708(89)90070-4 - A. O. L. Atkin and P. Swinnerton-Dyer,
*Some properties of partitions*, Proc. London Math. Soc. (3)**4**(1954), 84â€“106. MR**60535**, DOI 10.1112/plms/s3-4.1.84 - Alexander Berkovich and Frank G. Garvan,
*Some observations on Dysonâ€™s new symmetries of partitions*, J. Combin. Theory Ser. A**100**(2002), no.Â 1, 61â€“93. MR**1932070**, DOI 10.1006/jcta.2002.3281 - Kathrin Bringmann and Ken Ono,
*The $f(q)$ mock theta function conjecture and partition ranks*, Invent. Math.**165**(2006), no.Â 2, 243â€“266. MR**2231957**, DOI 10.1007/s00222-005-0493-5
BO2 K. Bringmann and K. Ono, - Youn-Seo Choi,
*Tenth order mock theta functions in Ramanujanâ€™s lost notebook*, Invent. Math.**136**(1999), no.Â 3, 497â€“569. MR**1695205**, DOI 10.1007/s002220050318 - Youn-Seo Choi,
*Tenth order mock theta functions in Ramanujanâ€™s lost notebook. II*, Adv. Math.**156**(2000), no.Â 2, 180â€“285. MR**1808245**, DOI 10.1006/aima.2000.1948
Dyson F. Dyson, - George Gasper and Mizan Rahman,
*Basic hypergeometric series*, Encyclopedia of Mathematics and its Applications, vol. 35, Cambridge University Press, Cambridge, 1990. With a foreword by Richard Askey. MR**1052153** - Dean Hickerson,
*A proof of the mock theta conjectures*, Invent. Math.**94**(1988), no.Â 3, 639â€“660. MR**969247**, DOI 10.1007/BF01394279 - Dean Hickerson,
*A proof of the mock theta conjectures*, Invent. Math.**94**(1988), no.Â 3, 639â€“660. MR**969247**, DOI 10.1007/BF01394279 - Victor G. Kac,
*Infinite-dimensional Lie algebras*, 3rd ed., Cambridge University Press, Cambridge, 1990. MR**1104219**, DOI 10.1017/CBO9780511626234 - Neal Koblitz,
*Introduction to elliptic curves and modular forms*, 2nd ed., Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1993. MR**1216136**, DOI 10.1007/978-1-4612-0909-6 - James Lepowsky and Robert Lee Wilson,
*A new family of algebras underlying the Rogers-Ramanujan identities and generalizations*, Proc. Nat. Acad. Sci. U.S.A.**78**(1981), no.Â 12, 7254â€“7258. MR**638674**, DOI 10.1073/pnas.78.12.7254 - Jeremy Lovejoy,
*Rank and conjugation for the Frobenius representation of an overpartition*, Ann. Comb.**9**(2005), no.Â 3, 321â€“334. MR**2176595**, DOI 10.1007/s00026-005-0260-8
mcintosh R. McIntosh, - Goro Shimura,
*On modular forms of half integral weight*, Ann. of Math. (2)**97**(1973), 440â€“481. MR**332663**, DOI 10.2307/1970831
watson G. N. Watson, - Hamza Yesilyurt,
*Four identities related to third order mock theta functions in Ramanujanâ€™s lost notebook*, Adv. Math.**190**(2005), no.Â 2, 278â€“299. MR**2102658**, DOI 10.1016/j.aim.2003.12.007
ZwegersPhD S. P. Zwegers,

*Dysonâ€™s ranks and Maass forms*, Ann. of Math., in press.

*Some guesses in the theory of partitions*, Eureka (Cambridge)

**8**(1944), pages 10-15.

*H and K Family of Mock Theta Functions*, preprint.

*The mock theta functions (2),*Proc. London Math. Soc. (2)

**42**(1937), pages 274-304.

*Mock theta functions*, Ph.D. Thesis, Universiteit Utrecht, 2002.

## Bibliographic Information

**Kathrin Bringmann**- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- MR Author ID: 774752
- Email: bringman@math.umn.edu
**Ken Ono**- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- MR Author ID: 342109
- Email: ono@math.wisc.edu
**Robert C. Rhoades**- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- MR Author ID: 762187
- Email: rhoades@math.wisc.edu
- Received by editor(s): March 28, 2007
- Published electronically: November 26, 2007
- Additional Notes: The authors thank the National Science Foundation for its support. The third author is supported by an NSF Graduate Fellowship and a National Physical Science Consortium Graduate Fellowship.
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**21**(2008), 1085-1104 - MSC (2000): Primary 11P99, 11F11, 05A19
- DOI: https://doi.org/10.1090/S0894-0347-07-00587-5
- MathSciNet review: 2425181