Proofs of conjectures of Sandon and Zanello on colored partition identities

Bruce C.~Berndt; Roberta R.~Zhou

doi:10.4134/JKMS.2014.51.5.987

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2014; 51(5): 987-1028

Printed September 1, 2014

https://doi.org/10.4134/JKMS.2014.51.5.987

Proofs of conjectures of Sandon and Zanello on colored partition identities

Bruce C.~Berndt and Roberta R.~Zhou

University of Illinois, Dalian University of Technology

Abstract

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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

MSC numbers: Primary 11P84; Secondary 05A15, 05A17