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

ABSTRACT

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.

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.

Chen, S. (2014) Symmetric Identities from an Invariant in Partition Conjugation and Their Applications in

References

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[3] Liu, Z.G. (2003) Some Operator Identities and Q-Series Transformation Formulas. Discrete Mathematics, 265, 119-139.

http://dx.doi.org/10.1016/S0012-365X(02)00626-X.

[4] Fine, N.J. (1988) Basic Hypergeometric Series and Applications. Mathematical Surveys and Monographs, 1988.

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[7] Andrews, G.E. (1966) On Basic Hypergeometric Series, Mock Theta Functions, and Partitions (I). The Quarterly Journal of Mathematics, 17, 64-80.

http://dx.doi.org/10.1093/qmath/17.1.64.

[8] Watson, G.N. (1936) The Final Problem: An Account of the Mock Theta Functions. Journal of the London Mathematical Society, 11, 55-80.

http://dx.doi.org/10.1112/jlms/s1-11.1.55

[9] Liu, X.C. (2012) On Flushed Partitions and Concave Compositions. European Journal of Combinatorics, 33, 663-678.

http://dx.doi.org/10.1016/j.ejc.2011.12.004

[10] Ramanujan, S. (1988) The Lost Notebook and Other Unpublished Paper.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

http://dx.doi.org/10.1007/s11139-006-0241-5

[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.

http://dx.doi.org/10.1007/s00026-003-0194-y.

[14] Warnaar, S.O. (2003) Partial Theta Functions. I. Beyond the Lost Notebook. Proceedings of the London Mathematical Society, 87, 363-395.

http://dx.doi.org/10.1112/S002461150201403X.

[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.

http://dx.doi.org/10.2307/2321943

[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.

http://dx.doi.org/10.1016/S0012-365X(02)00626-X.

[4] Fine, N.J. (1988) Basic Hypergeometric Series and Applications. Mathematical Surveys and Monographs, 1988.

http://dx.doi.org/10.1090/surv/027

[5] Andrews, G.E. (1972) Two Theorems of Gauss and Allied Identities Proved Arithmetically. Pacific Journal of Mathematics, 41, 563-578.

http://dx.doi.org/10.2140/pjm.1972.41.563.

[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.

http://dx.doi.org/10.1093/qmath/17.1.64.

[8] Watson, G.N. (1936) The Final Problem: An Account of the Mock Theta Functions. Journal of the London Mathematical Society, 11, 55-80.

http://dx.doi.org/10.1112/jlms/s1-11.1.55

[9] Liu, X.C. (2012) On Flushed Partitions and Concave Compositions. European Journal of Combinatorics, 33, 663-678.

http://dx.doi.org/10.1016/j.ejc.2011.12.004

[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.

http://dx.doi.org/10.1007/s11139-006-0241-5

[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.

http://dx.doi.org/10.1007/s00026-003-0194-y.

[14] Warnaar, S.O. (2003) Partial Theta Functions. I. Beyond the Lost Notebook. Proceedings of the London Mathematical Society, 87, 363-395.

http://dx.doi.org/10.1112/S002461150201403X.

[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.

http://dx.doi.org/10.2307/2321943