On combinatorial extensions of Rogers-Ramanujan type identities

Authors

DOI:

https://doi.org/10.55016/ojs/cdm.v12i2.62496

Keywords:

\((n t)\)-color partitions, , anti-hook differences, combinatorial identities

Abstract

In the present paper we use anti-hook differences of Agarwal and Andrews as an elementary tool to provide new partition theoretic meanings to two generalized basic series in terms of ordinary partitions satisfying certain anti-hook difference conditions. Five particular cases are also discussed. These particular cases yield new partition theoretic versions of G\"{o}llnitz-Gordon identities and G\"{o}llnitz identity. Five $q$-identities of Rogers and three $q$-identities of Slater are further explored. These results extend the work of Goyal and Agarwal, Agarwal and Rana and Sareen and Rana.

Author Biography

  • Megha Goyal, Punjabi University, Patiala

    Assistant Professor,

    Basic and Applied Sciences,

    Punjabi University,

    Patiala-147002, India

     

References

A.K. Agarwal and G.E. Andrews, Hook differences and lattice paths, J. Statist. Plann. Inference, 14(1) (1986), 5--14.

A.K. Agarwal and G.E. Andrews, Rogers-Ramanujan identities for partitions with ``$n$ copies of $n$'', J. Combin. Theory, Ser. A,
45(1) (1987), 40--49.

A.K. Agarwal and D.M. Bressoud, Lattice paths and multiple basic
hypergeometric series, Pacific J. Math., 136(2) (1989), 209--228.

Ashok Kumar Agarwal and Megha Goyal, Lattice paths and Rogers
identities, Open J. of Discrete Mathematics, 1 (2011), 89--95.

A.K. Agarwal and Megha Goyal, On 3--way Combinatorial Identities, Proc. Indian Acad. Sci. (Math. Sci.), to appear.

A.K. Agarwal and M. Rana, New combinatorial versions of G\"{o}llnitz-Gordon identities, Utilitas Mathematica, 79 (2009), 145--155.

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H. G\"{o}llnitz, Einfache partitionen (unpublished), Diplomarbeit W.S., Gotttingen, 65 (1960).

H. G\"{o}llnitz, Partitionen unit differenzenbedingun-gen, J. Reine Angew. Math., 225 (1967), 154--190.

B. Gordon, Some continued fractions of the Rogers-Ramanujan type, Duke J. Math., 32 (1965), 741--748.

Megha Goyal, New combinatorial interpretations of some Rogers-Ramanujan type identities, Contrib. Discrete Math., to appear.

M. Goyal and A.K. Agarwal, Further Rogers-Ramanujan identities for
$n$-color partitions, Utilitas Mathematica, 95 (2014), 141--148.

M. Goyal and A.K. Agarwal, On a new class of combinatorial identities, ARS Combinatoria, (to appear).

J.K. Sareen and M. Rana, Four-way combinatorial interpretations of some
Rogers-Ramanujan type identities, ARS Combinatoria, (to appear).

L.J. Slater, Further identities of the Rogers-Ramanujan type, Proc.
London Math. Soc., 54(2) (1952), 147--167.

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Published

2017-11-27

Issue

Vol. 12 No. 2 (2017)

Section

Articles

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