Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T08:35:19.887Z Has data issue: false hasContentIssue false
  • English
  • Français

Taylor coefficients of mock-Jacobi forms and moments of partition statistics

Published online by Cambridge University Press:  09 July 2014

KATHRIN BRINGMANN ,
KARL MAHLBURG  and
ROBERT C. RHOADES
KATHRIN BRINGMANN
Affiliation:
Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany e-mail [email protected]
KARL MAHLBURG
Affiliation:
Department of Mathematics, Princeton University, NJ, U.S.A. e-mail [email protected]
ROBERT C. RHOADES
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, U.S.A. e-mail [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a new technique for deriving asymptotic series expansions for moments of combinatorial generating functions that uses the transformation theory of Jacobi forms and “mock” Jacobi forms, as well as the Hardy-Ramanujan Circle Method. The approach builds on a suggestion of Zagier, who observed that the moments of a combinatorial statistic can be simultaneously encoded as the Taylor coefficients of a function that transforms as a Jacobi form. Our use of Jacobi transformations is a novel development in the subject, as previous results on the asymptotic behavior of the Taylor coefficients of Jacobi forms have involved the study of each such coefficient individually using the theory of quasimodular forms and quasimock modular forms.

As an application, we find asymptotic series for the moments of the partition rank and crank statistics. Although the coefficients are exponentially large, the error in the series expansions is polynomial, and have the same order as the coefficients of the residual Eisenstein series that are undetectable by the Circle Method. We also prove asymptotic series expansions for the symmetrized rank and crank moments introduced by Andrews and Garvan, respectively. Equivalently, the former gives asymptotic series for the enumeration of Andrews k-marked Durfee symbols.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 2 , September 2014 , pp. 231 - 251
Copyright
Copyright © Cambridge Philosophical Society 2014 

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.Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks. Invent. Math. 169 (2007), 37–73.CrossRefGoogle Scholar
[2]Andrews, G.The number of smallest parts in the partitions of n. J. Reine Angew. Math. 624 (2008), 133–142.Google Scholar
[3]Andrews, G.The Theory of Partitions. Cambridge University Press, Cambridge, 1998.Google Scholar
[4]Andrews, G. and Garvan, F.Dyson's crank of a partition. Bull. Amer. Math. Soc. 18 (1988), 167–171.CrossRefGoogle Scholar
[5]Atkin, A. and Garvan, F.Relations between the ranks and the cranks of partitions. Ramanujan J. 7 (2003), 137–152.CrossRefGoogle Scholar
[6]Atkin, A. and Swinnerton-Dyer, H.Some properties of partitions. Proc. London Math. Soc. 4 (1954), 84–106.CrossRefGoogle Scholar
[7]Bringmann, K.On the construction of higher deformations of partition statistics. Duke Math. J. 144 (2008), 195–233.CrossRefGoogle Scholar
[8]Bringmann, K. and Folsom, A.On the asymptotic behavior of Kac–Wakimoto characters. Proc. Amer. Math. Soc. 141 (2013), 1567–1576.CrossRefGoogle Scholar
[9]Bringmann, K., Garvan, F. and Mahlburg, K. Partition statistics and quasiweak Maass forms. Int. Math. Res. Not. (2009), no. 1, 63–97.Google Scholar
[10]Bringmann, K. and Mahlburg, K.Inequalities between ranks and cranks. Proc. Amer. Math. Soc. 137 (2009), 541–552.CrossRefGoogle Scholar
[11]Bringmann, K. and Mahlburg, K.Asymptotic formulas for coefficients of Kac-Wakimoto characters. Math. Proc. Camb. Phil. Soc. 155 (2013), 51–72.CrossRefGoogle Scholar
[12]Bringmann, K., Mahlburg, K. and Rhoades, R.Asymptotics for rank and crank moments. Bull. London Math. Soc. 43 (2011), 661–672.CrossRefGoogle Scholar
[13]Bringmann, K. and Ono, K.Dyson's ranks and Maass forms. Ann. of Math. 171 (2010), 419–449.CrossRefGoogle Scholar
[14]Bringmann, K. and Richter, O.Zagier-type dualities and lifting maps for harmonic weak Maass–Jacobi form. Adv. of Math. 225 (2010), 2298–2315.CrossRefGoogle Scholar
[15]Bringmann, K. and Zwegers, S.Rank-crank type PDE's and non-holomorphic Jacobi forms. Math. Res. Lett. 17 (2010), 589–600.CrossRefGoogle Scholar
[16]Bruinier, J. and Funke, J.On two geometric theta lifts. Duke Math. J. 125 (2004), 45–90.CrossRefGoogle Scholar
[17]Dyson, F.Some guesses in the theory of partitions. Eureka (Cambridge) 8 (1944), 10–15.Google Scholar
[18]Dyson, F.Mappings and symmetries of partitions. J. Combin. Theory Ser. A 51 (1989), 169–180.CrossRefGoogle Scholar
[19]Eichler, M. and Zagier, D.The theory of Jacobi forms.Prog. Math. 55. (Birkhäuser Boston, Inc., Boston, MA, 1985).Google Scholar
[20]Garvan, F.New combinatorial interpretations of Ramanujan's partition congruences mod 5, 7 and 11. Trans. Amer. Math. Soc. 305 (1988), 47–77.Google Scholar
[21]Garvan, F.Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank. Int. J. Number Theory 6 (2010), 1–29.CrossRefGoogle Scholar
[22]Garvan, F.Higher Order spt-Functions. Adv. in Math. 228 (2011), 241–265.CrossRefGoogle Scholar
[23]Lehner, J.On automorphic forms of negative dimension. Illinois J. Math. 8 (1964), 395–407.CrossRefGoogle Scholar
[24]Mahlburg, K.Partition congruences and the Andrews–Garvan-Dyson crank. Proc. Natl. Acad. Sci. 102 (2005), no. 43, 15373–15376.CrossRefGoogle ScholarPubMed
[25]Ramanujan, S.Congruence properties of partitions. Math. Zeit. 9 (1921), 147–153.CrossRefGoogle Scholar
[26]Shimura, G.On modular forms of half integral weight. Ann. of Math. (2) 97 (1973), 440–481.CrossRefGoogle Scholar
[27]Zagier, D. Ramanujan's mock theta functions and their applications (after Zwegers and Ono-Bringmann). Séminaire Bourbaki. Vol. 2007/2008. Astérisque No. 326 (2009), Exp. No. 986 (2010), vii–viii 143–164.Google Scholar
[28]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
[29]Zwegers, S. Mock theta functions. PhD. Thesis. Universiteit Utrecht (2002).Google Scholar
[30]Zwegers, S.Rank-Crank type PDE's for higher level Appell functions. Acta Arith. 144 (2010), 263–273CrossRefGoogle Scholar