PROOFS OF CONJECTURES OF SANDON AND ZANELLO ON COLORED PARTITION IDENTITIES

- Journal title : Journal of the Korean Mathematical Society
- Volume 51, Issue 5, 2014, pp.987-1028
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2014.51.5.987

Title & Authors

PROOFS OF CONJECTURES OF SANDON AND ZANELLO ON COLORED PARTITION IDENTITIES

Berndt, Bruce C.; Zhou, Roberta R.;

Berndt, Bruce C.; Zhou, Roberta R.;

Abstract

In a recent systematic study, C. Sandon and F. Zanello offered 30 conjectured identities for partitions. As a consequence of their study of partition identities arising from Ramanujan's formulas for multipliers in the theory of modular equations, the present authors in an earlier paper proved three of these conjectures. In this paper, we provide proofs for the remaining 27 conjectures of Sandon and Zanello. Most of our proofs depend upon known modular equations and formulas of Ramanujan for theta functions, while for the remainder of our proofs it was necessary to derive new modular equations and to employ the process of duplication to extend Ramanujan's catalogue of theta function formulas.

Keywords

colored partitions;modular equations;theta function identities;

Language

English

Cited by

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References

1.

G. E. Andrews, The Theory of Partitions, Encycl. Math. and Its Appl. Vol. 2, Addison-Wesley, Reading, 1976; reissued by Cambridge University Press, Cambridge, 1998.

2.

G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook. Part I, Springer, New York, 2005.

3.

G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge University Press, Cambridge, 2004.

4.

N. D. Baruah and B. C. Berndt, Partition identities and Ramanujan's modular equations, J. Combin. Theory Ser. A 114 (2007), no. 6, 1024-1045.

5.

N. D. Baruah and B. C. Berndt, Partition identities arising from theta function identities, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 6, 955-970.

6.

N. D. Baruah and B. Boruah, Colored partition identities conjectured by Sandon and Zanello, Ramanujan J., to appear.

7.

B. C. Berndt, Ramanujan's Notebooks. Part III, Springer-Verlag, New York, 1991.

8.

B. C. Berndt, Number Theory in the Spirit of Ramanujan, American Mathematical Society, Providence, RI, 2006.

9.

B. C. Berndt, Partition-theoretic interpretations of certain modular equations of Schroter, Russell, and Ramanujan, Ann. Comb. 11 (2007), no. 2, 115-125.

10.

B. C. Berndt and R. R. Zhou, Identities for partitions with distinct colors, Ann. Combin., to appear.

11.

W. Chu and L. Di Claudio, Classical Partition Identities and Basic Hypergeometric Series, Quaderno 6/2004 del Dipartimento di Matematica "Ennio De Giorgi", Universit'a degli Studi di Lecce, 2004, Edizioni del Grifo, Lecce.

12.

H. M. Farkas and I. Kra, Partitions and theta constant identities, in The Mathematics of Leon Ehrenpreis, Contemp. Math. No. 251, pp. 197-203, American Mathematical Society, Providence, RI, 2000.

13.

S. Kim, Bijective proofs of partition identities arising from modular equations, J. Combin. Theory Ser. A 116 (2009), no. 3, 699-712.

14.

S. Ramanujan, Notebooks, (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.

15.

C. Sandon and F. Zanello, Warnaar's bijection and colored partition identities. I, J. Combin. Theory Ser. A 120 (2013), no. 1, 28-38.