On 3-way combinatorial identities
Article
First Online:
Received:
Revised:
Accepted:
- 15 Downloads
Abstract
In this paper, we provide combinatorial meanings to two generalized basic series with the aid of associated lattice paths. These results produce two new classes of infinite 3-way combinatorial identities. Five particular cases are also discussed. These particular cases provide new combinatorial versions of Göllnitz–Gordon identities and Göllnitz identity. Seven q-identities of Slater and five q-identities of Rogers are further explored using the same combinatorial object. These results are an extension of the work of Goyal and Agarwal (Utilitas Math. 95 (2014) 141–148), Agarwal and Rana (Utilitas Math. 79 (2009) 145–155), and Agarwal (J. Number Theory 28 (1988) 299–305).
Keywords
Basic series n-color partitions lattice paths associated lattice paths combinatorial identities2010 Mathematics Subject Classification
05A15 05A17 11P81References
- 1.Agarwal A K, Rogers–Ramanujan identities for \(n\)-color partitions, J. Number Theory 28 (1988) 299–305MathSciNetCrossRefMATHGoogle Scholar
- 2.Agarwal A K, Lattice paths and \(n\)-color partitions, Utilitas Mathematica 53 (1998) 71–80MathSciNetMATHGoogle Scholar
- 3.Agarwal A K and Andrews G E, Rogers–Ramanujan identities for partitions with ‘\(N\) copies of \(N\)’, J. Combin. Theory Ser. A. 45(1) (1987) 40–49MathSciNetCrossRefMATHGoogle Scholar
- 4.Agarwal A K and Bressoud D M, Lattice paths and multiple basic hypergeometric series, Pac. J. Math. 136(2) (1989) 209–228MathSciNetCrossRefMATHGoogle Scholar
- 5.Agarwal A K and Goyal M, Lattice paths and Rogers identities, Open J. Discrete Math. 1 (2011) 89–95MathSciNetCrossRefMATHGoogle Scholar
- 6.Agarwal A K and Goyal M, New partition theoretic interpretations of Rogers–Ramanujan identities, Int. J. Combin. (2012) 6 pages, https://doi.org/10.1155/2012/409505
- 7.Agarwal A K and Rana M, New combinatorial versions of Göllnitz–Gordon identities, Utilitas Mathematica 79 (2009) 145–155MathSciNetMATHGoogle Scholar
- 8.Agarwal A K and Rana M, On an extension of a combinatorial identity, Proc. Indian Acad. Sci. (Math. Sci.) 119(1) (2009) 1–7MathSciNetCrossRefMATHGoogle Scholar
- 9.Anand S and Agarwal A K, A new class of lattice paths and partitions with \(n\) copies of \(n\), Proc. Indian Acad. Sci. (Math. Sci.) 122(1) (2012) 23–39MathSciNetCrossRefMATHGoogle Scholar
- 10.Andrews G E, An introduction to Ramanujan’s “lost” notebook, Am. Math. Mon. 86 (1979) 89–108MathSciNetCrossRefMATHGoogle Scholar
- 11.Connor W G, Partition theorems related to some identities of Rogers and Watson, Trans. Am. Math. Soc. 214 (1975) 95–111MathSciNetCrossRefMATHGoogle Scholar
- 12.Ganesan Ghurumuruhan, Multiplicity of summands in the random partitions of an integer, Proc. Indian Acad. Sci. (Math. Sci.) 123(1) (2013) 101–143MathSciNetCrossRefMATHGoogle Scholar
- 13.Göllnitz H, Einfache partitionen (unpublished), Diplomarbeit W.S., Gotttingen (1960) 65Google Scholar
- 14.Göllnitz H, Partitionen unit differenzenbedingun-gen, J. Reine Angew. Math. 225 (1967) 154–190MathSciNetMATHGoogle Scholar
- 15.Gordon B, Some continued fractions of the Rogers–Ramanujan type, Duke J. Math. 32 (1965) 741–748MathSciNetCrossRefMATHGoogle Scholar
- 16.Goyal M and Agarwal A K, Further Rogers–Ramanujan identities for \(n\)-color partitions, Utilitas Mathematica 95 (2014) 141–148MathSciNetMATHGoogle Scholar
- 17.Goyal M and Agarwal A K, On a new class of combinatorial identities, ARS Combin. 127 (2016) 65–77MathSciNetMATHGoogle Scholar
- 18.Mansour Toufik and Shattuck Mark, A statistic on \(n\)-color compositions and related sequences, Proc. Indian Acad. Sci. (Math. Sci.) 124(2) (2014) 127–140MathSciNetCrossRefMATHGoogle Scholar
- 19.Slater L J, Further identities of the Rogers–Ramanujan type, Proc. Lond. Math. Soc. 54 (1952) 147–167MathSciNetCrossRefMATHGoogle Scholar
- 20.Subbarao M V, Some Rogers–Ramanujan type partition theorems, Pac. J. Math. 120 (1985) 431–435MathSciNetCrossRefMATHGoogle Scholar
Copyright information
© Indian Academy of Sciences 2018