Abstract
We show how Rank–Crank-type PDEs for higher order Appell functions due to Zwegers may be obtained from a generalized Lambert series identity due to the first author. Special cases are the Rank–Crank PDE due to Atkin and the third author and a PDE for a level 5 Appell function also found by the third author. These two special PDEs are related to generalized Lambert series identities due to Watson, and Jackson, respectively. The first author’s Lambert series identity is a common generalization. We also show how Atkin and Swinnerton-Dyer’s proof using elliptic functions can be extended to prove these generalized Lambert series identities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrews, G.E., Garvan, F.G.: Dyson’s crank of a partition. Bull., New Ser., Am. Math. Soc. 18, 167–171 (1988)
Atkin, A.O.L., Garvan, F.G.: Relations between the ranks and cranks of partitions. Ramanujan J. 7, 343–366 (2003)
Atkin, A.O.L., Swinnerton-Dyer, H.P.F.: Some properties of partitions. Proc. Lond. Math. Soc. 4(3), 84–106 (1954)
Berndt, B.C.: Number Theory in the Spirit of Ramanujan. Am. Math. Soc., Providence (2006)
Bringmann, K., Lovejoy, J., Osburn, R.: Rank and crank moments for overpartitions. J. Number Theory 129, 1758–1772 (2009)
Bringmann, K., Lovejoy, J., Osburn, R.: Automorphic properties of generating functions for generalized rank moments and Durfee symbols. Int. Math. Res. Not. 2, 238–260 (2010)
Bringmann, K., Zwegers, S.: Rank–Crank type PDE’s and non-holomorphic Jacobi forms. Math. Res. Lett. 17, 589–600 (2010)
Chan, S.H.: Generalized Lambert series identities. Proc. Lond. Math. Soc. (3) 91, 598–622 (2005)
Dyson, F.J.: Some guesses in the theory of partitions. Eureka (Camb.) 8, 10–15 (1944)
Dyson, F.J.: Selected Papers of Freeman Dyson with Commentary. Am. Math. Soc., Providence (1996)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. II. McGraw-Hill, New York (1953)
Garvan, F.G.: New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11. Trans. Am. Math. Soc. 305, 47–77 (1988)
Garvan, F.G.: Generalizations of Dyson’s ranks and non-Rogers-Ramanujan partitions. Manuscr. Math. 84, 343–359 (1994)
Jackson, M.: On some formulae in partition theory and bilateral basic hypergeometric series. J. Lond. Math. Soc. 24, 233–237 (1949)
Lewis, R.P.: Relations between ranks and cranks modulo 9. J. Lond. Math. Soc. 45(2), 222–231 (1992)
Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi (1988)
Tannery, J., Molk, J.: Éléments de la Théorie des Fonctions Elliptiques, vol. III. Gauthier-Villars, Paris (1896). (Reprinted: Chelsea, New York, 1972)
Watson, G.N.: The final problem: An account of the mock theta functions. J. Lond. Math. Soc. 11, 55–80 (1936)
Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1966)
Zwegers, S.P.: Mock theta functions. Ph.D. Thesis, Universiteit Utrecht (2002)
Zwegers, S.P.: Rank–Crank type PDE’s for higher level Appell functions. Acta Arith. 144, 263–273 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to our friends, Mourad Ismail and Dennis Stanton
The first author was partially supported by Nanyang Technological University Academic Research Fund, Project Number RG68/10.
The third author was supported in part by NSA Grant H98230-09-1-0051.
Rights and permissions
About this article
Cite this article
Chan, S.H., Dixit, A. & Garvan, F.G. Rank–Crank-type PDEs and generalized Lambert series identities. Ramanujan J 31, 163–189 (2013). https://doi.org/10.1007/s11139-012-9373-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-012-9373-y
Keywords
- Rank–Crank PDE
- Higher level Appell function
- Lambert series
- Partition
- q-series
- Basic hypergeometric function
- Quasimodular form