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


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.

Share and Cite:

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.

Conflicts of Interest

The authors declare no conflicts of interest.


[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[10] Ramanujan, S. (1988) The Lost Notebook and Other Unpublished Paper.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.

Copyright © 2023 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.