×

zbMATH — the first resource for mathematics

Partition identities and Ramanujan’s modular equations. (English) Zbl 1206.11132
Summary: We show that certain modular equations and theta function identities of Ramanujan imply elegant partition identities. Several of the identities are for \(t\)-cores.

MSC:
11P84 Partition identities; identities of Rogers-Ramanujan type
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adiga, C.; Berndt, B.C.; Bhargava, S.; Watson, G.N., Chapter 16 of Ramanujan’s second notebook: theta-functions and q-series, Mem. amer. math. soc., 53, 315, (1985) · Zbl 0565.33002
[2] Andrews, G.E.; Berndt, B.C., Ramanujan’s lost notebook, part I, (2005), Springer New York · Zbl 1075.11001
[3] N.D. Baruah, B.C. Berndt, Partition identities arising from theta function identities, submitted for publication · Zbl 1154.11036
[4] Berndt, B.C., Ramanujan’s notebooks, part III, (1991), Springer-Verlag New York · Zbl 0733.11001
[5] Berndt, B.C., Ramanujan’s notebooks, part IV, (1994), Springer-Verlag New York · Zbl 0785.11001
[6] Berndt, B.C., Ramanujan’s notebooks, part V, (1998), Springer-Verlag New York · Zbl 0886.11001
[7] B.C. Berndt, Partition-theoretic interpretations of certain modular equations of Schröter, Russell, and Ramanujan, Ann. Comb., in press
[8] Farkas, H.M.; Kra, I., Partitions and theta constant identities, (), 197-203 · Zbl 1050.11086
[9] Farkas, H.M.; Kra, I., Theta constants, Riemann surfaces and the modular group, Grad. stud. math., vol. 37, (2001), Amer. Math. Soc. Providence, RI · Zbl 1053.11031
[10] Garvan, F.; Kim, D.; Stanton, D., Cranks and t-cores, Invent. math., 101, 1-17, (1990) · Zbl 0721.11039
[11] Granville, A.; Ono, K., Defect zero p-blocks for finite simple groups, Trans. amer. math. soc., 348, 331-347, (1996) · Zbl 0855.20007
[12] Hirschhorn, M.D., The case of the mysterious sevens, Int. J. number theory, 2, 213-216, (2006) · Zbl 1116.11083
[13] Ramanujan, S., Notebooks (2 volumes), (1957), Tata Institute of Fundamental Research Bombay
[14] Ramanujan, S., The lost notebook and other unpublished papers, (1988), Narosa New Delhi · Zbl 0639.01023
[15] H. Schröter, De aequationibus modularibus, Dissertatio Inauguralis, Albertina Litterarum Universitate, Regiomonti, 1854
[16] Schröter, H., Über modulargleichungen der elliptischen funktionen, auszug aus einem schreiben an herrn L. Kronecker, J. reine angew. math., 58, 378-379, (1861)
[17] Schröter, H., Beiträge zur theorie der elliptischen funktionen, Acta math., 5, 205-208, (1884) · JFM 16.0409.01
[18] J.A. Sellers, An elementary proof of an infinite family of identities for 3-cores, preprint, May 10, 2006
[19] Warnaar, S.O., A generalization of the farkas and kra partition theorem for modulus 7, J. combin. theory ser. A, 110, 43-52, (2005) · Zbl 1101.11046
[20] Weber, H., Zur theorie der elliptischen funktionen, Acta math., 11, 333-390, (1887) · JFM 20.0471.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.