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Contributions of Alladi Ramakrishnan to the Mathematical Sciences Read chapter Read first chapter

2010 | OriginalPaper | Chapter

Partitions with Non-Repeating Odd Parts and Q-Hypergeometric Identities

Author : Krishnaswami Alladi

Published in: The Legacy of Alladi Ramakrishnan in the Mathematical Sciences

Publisher: Springer New York

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Summary

We obtain a series expansion for the product generating function of partitions in which the odd parts do not repeat. This is done by studying the 2-modular Ferrers graphs of such partitions via Durfee squares. This provides a unified approach to several fundamental identities in the theory of partitions and q-series such as those of Sylvester, Lebesgue, Gauss, and Rogers-Fine, and provides links with Göllnitz’s deep theorem.

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previous chapter Inversion and Invariance of Characteristic Terms: Part I
next chapter q-Catalan Identities
Footnotes
1
It is worthwhile to note that under dilations (in this case q↦q 2) and translations, the underlying combinatorics can change and this is non-trivial; in going from Theorem 1 to Theorem 2, the surjection changed to a bijection.
 
Literature
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K. Alladi, “A variation on a theme of Sylvester – a smoother road to Göllnitz’s partition theorem”, Discrete Math. 196 (1999), 1–11.MathSciNetMATHCrossRef
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K. Alladi, “A new combinatorial study of the Rogers-Fine identity and a related partial theta series”, Int’l J. Num. Th., 5 (2009), 1311–1320.MathSciNetMATHCrossRef
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K. Alladi, G. E. Andrews and B. Gordon, “Generalizations and refinements of a partition theorem of Göllnitz” J. Reine Ang. Math., 460 (1995), 165–188.MathSciNetMATH
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M. Hirschhorn and J. Sellers, “Arithmetic properties of partitions with odd parts distinct”, Ramanujan J., 22 (2010, to appear).
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M. J. Rowell and A. J. Yee, “A bijective proof of a limiting case Watson’s 8ϕ7 transformation formula”, Ramanujan J., 20 (2009), 267–280.MathSciNetCrossRef
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Metadata
Title
Partitions with Non-Repeating Odd Parts and Q-Hypergeometric Identities
Author
Krishnaswami Alladi
Copyright Year
2010
Publisher
Springer New York
Book
The Legacy of Alladi Ramakrishnan in the Mathematical Sciences

Print ISBN: 978-1-4419-6262-1

Electronic ISBN: 978-1-4419-6263-8

Copyright Year: 2010

DOI
https://doi.org/10.1007/978-1-4419-6263-8_9

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