Share This Article:

Symmetric Identities from an Invariant in Partition Conjugation and Their Applications in q-Series

Abstract Full-Text HTML Download Download as PDF (Size:276KB) PP. 36-43
DOI: 10.4236/ojdm.2014.42006    4,308 Downloads   5,499 Views  


For every partition and its conjugation , there is an important invariant , which denotes the number of different parts. That is , . We will derive a series of symmetric q-identities from the invariant in partition conjugation by studying modified Durfee rectangles. The extensive applications of the several symmetric q-identities in q-series  [1] will also be discussed. Without too much effort one can obtain much well-known knowledge as well as new formulas by proper substitutions and elementary calculations, such as symmetric identities, mock theta functions, a two-variable reciprocity theorem, identities from Ramanujan’s Lost Notebook and so on.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Chen, S. (2014) Symmetric Identities from an Invariant in Partition Conjugation and Their Applications in q-Series. Open Journal of Discrete Mathematics, 4, 36-43. doi: 10.4236/ojdm.2014.42006.


[1] Gasper, G. and Rahman, M. (2004) Basic Hypergeometric Series. 2nd Edition, Cambridge University Press, Cambridge.
[2] Andrews, G.E. (1976) The Theory of Partitions, Encyclopedia of Math, and Its Applications. Addison-Wesley Publishing Co., Boston.
[3] Liu, Z.G. (2003) Some Operator Identities and Q-Series Transformation Formulas. Discrete Mathematics, 265, 119-139.
[4] Fine, N.J. (1988) Basic Hypergeometric Series and Applications. Mathematical Surveys and Monographs, 1988.
[5] Andrews, G.E. (1972) Two Theorems of Gauss and Allied Identities Proved Arithmetically. Pacific Journal of Mathematics, 41, 563-578.
[6] Berndt, B.C. and Rankin, R.A. (1995) Ramanujan: Letters and Commentary. American Mathematical Society, Providence, London Mathematical Society, London.
[7] Andrews, G.E. (1966) On Basic Hypergeometric Series, Mock Theta Functions, and Partitions (I). The Quarterly Journal of Mathematics, 17, 64-80.
[8] Watson, G.N. (1936) The Final Problem: An Account of the Mock Theta Functions. Journal of the London Mathematical Society, 11, 55-80.
[9] Liu, X.C. (2012) On Flushed Partitions and Concave Compositions. European Journal of Combinatorics, 33, 663-678.
[10] Ramanujan, S. (1988) The Lost Notebook and Other Unpublished Paper. Springer-Verlag, Berlin.
[11] Berndt, B.C., Chan, S.H., Yeap, B.P. and Yee, A.J. (2007) A Reciprocity Theorem for Certain Q-Series Found in Ramanujan’s Lost Notebook. The Ramanujan Journal, 13, 27-37.
[12] Andrews, G.E. and Berndt, B.C. (2005) Ramanujan’s Lost Notebook, Part I. Springer, New York.
[13] Berndt, B.C. and Yee, A.J. (2003) Combinatorial Proofs of Identities in Ramanujan’s Lost Notebook Associated with the Rogers-Fine Identity and False Theta Functions. Annals of Combinatorics, 7, 409-423.
[14] Warnaar, S.O. (2003) Partial Theta Functions. I. Beyond the Lost Notebook. Proceedings of the London Mathematical Society, 87, 363-395.
[15] Rogers, L.J. (1917) On Two Theorems of Combinatory Analysis and Some Allied Identities. Proceedings of the London Mathematical Society, 16, 316-336.
[16] Andrews, G.E. (1979) An Introduction to Ramanujan’s Lost Notebook, The American Mathematical Monthly, 86, 89-108.

comments powered by Disqus

Copyright © 2019 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.