On combinatorial extensions of Rogers-Ramanujan type identities





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


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



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.

G.E. Andrews, An introduction to Ramanujan’s “LOST” notebook, Amer. Math. Monthly, 86 (1979), 89--108.

G.E. Andrews, Generalized Frobenius partitions, Mem. Amer. Math. Soc., 49(301) (1984), iv+44pp.

H. G\"{o}llnitz, Einfache partitionen (unpublished), Diplomarbeit W.S., Gotttingen, 65 (1960).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