Vector-valued Maass-Poincaré series

Author:
Sharon Anne Garthwaite

Journal:
Proc. Amer. Math. Soc. **136** (2008), 427-436

MSC (2000):
Primary 11F30; Secondary 11F37

DOI:
https://doi.org/10.1090/S0002-9939-07-08961-7

Published electronically:
November 1, 2007

MathSciNet review:
2358480

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Abstract | References | Similar Articles | Additional Information

Abstract: Shortly before his death, Ramanujan wrote about his discovery of mock theta functions, functions with interesting analytic properties. Recently, Zweger showed that mock theta functions could be ``completed'' to satisfy the transformation properties of a weight real analytic vector-valued modular form. Using Maass-Poincaré series, Bringmann and Ono proved the Andrews-Dragonette conjecture, establishing an exact formula for the coefficients of Ramanujan's mock theta function . In this paper we study vector-valued Maass-Poincaré series of all weights, and give their Fourier expansions.

**1.**George E. Andrews, Richard Askey, and Ranjan Roy,*Special functions*, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR**1688958****2.**Kathrin Bringmann and Ken Ono,*Arithmetic properties of coefficients of half-integral weight Maass-Poincaré series*, Mathematische Annalen,**337**(2007). 591-612.**3.**-,*Dyson's ranks and Maass forms*, accepted for publication in Annals of Mathematics.**4.**Kathrin Bringmann and Ken Ono,*The 𝑓(𝑞) mock theta function conjecture and partition ranks*, Invent. Math.**165**(2006), no. 2, 243–266. MR**2231957**, https://doi.org/10.1007/s00222-005-0493-5**5.**Jan H. Bruinier,*Borcherds products on O(2, 𝑙) and Chern classes of Heegner divisors*, Lecture Notes in Mathematics, vol. 1780, Springer-Verlag, Berlin, 2002. MR**1903920****6.**Freeman J. Dyson,*A walk through Ramanujan’s garden*, Ramanujan revisited (Urbana-Champaign, Ill., 1987) Academic Press, Boston, MA, 1988, pp. 7–28. MR**938957****7.**Sharon Anne Garthwaite,*The coefficients of the mock theta function*, Accepted for publication,*International Journal of Number Theory*.**8.**Dennis A. Hejhal,*The Selberg trace formula for 𝑃𝑆𝐿(2,𝑅). Vol. 2*, Lecture Notes in Mathematics, vol. 1001, Springer-Verlag, Berlin, 1983. MR**711197****9.**Srinivasa Ramanujan,*The lost notebook and other unpublished papers*, Springer-Verlag, Berlin; Narosa Publishing House, New Delhi, 1988. With an introduction by George E. Andrews. MR**947735****10.**S. P. Zwegers,*Mock 𝜃-functions and real analytic modular forms*, 𝑞-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000) Contemp. Math., vol. 291, Amer. Math. Soc., Providence, RI, 2001, pp. 269–277. MR**1874536**, https://doi.org/10.1090/conm/291/04907**11.**Sander Zwegers,*Mock theta functions*, Ph.D. thesis, Universiteit Utrecht, 2002.

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Additional Information

**Sharon Anne Garthwaite**

Affiliation:
Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837

Email:
sharon.garthwaite@bucknell.edu

DOI:
https://doi.org/10.1090/S0002-9939-07-08961-7

Received by editor(s):
April 16, 2006

Received by editor(s) in revised form:
October 17, 2006

Published electronically:
November 1, 2007

Additional Notes:
This research was supported by the University of Wisconsin at Madison NSF VIGRE program

Communicated by:
Ken Ono

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.