On a 3-Way Combinatorial Identity


Recently in [1] Goyal and Agarwal interpreted a generalized basic series as a generating function for a colour partition function and a weighted lattice path function. This led to an infinite family of combinatorial identities. Using Frobenius partitions, we in this paper extend the result of [1] and obtain an infinite family of 3-way combinatorial identities. We illustrate by an example that our main result has a potential of yielding Rogers-Ramanujan-MacMahon type identities with convolution property.

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Sood, G. and Agarwal, A. (2014) On a 3-Way Combinatorial Identity. Open Journal of Discrete Mathematics, 4, 89-96. doi: 10.4236/ojdm.2014.44012.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Goyal, M. and Agarwal, A.K. On a New Class of Combinatorial Identities. ARS Combinatoria. (to appear).
[2] Rogers, L.J. (1894) Second Memoir on the Expansion of Certain Infinite Products. Proceedings London Mathematical Society, 25, 318-343.
[3] MacMohan, P.A. (1916) Combinatory Analysis. Vol. 2, Cambridge University Press, London and New York.
[4] Göllnitz, H. (1960) Einfache Partitionen. Diplomarbeit W.S., Gotttingen, 65 p. (unpublished)
[5] Göllnitz, H. (1967) Partitionen mit Differenzenbedingungen. Journal für die Reine und Angewandte Mathematik, 225, 154-190.
[6] Gordon, B. (1965) Some Continued Fractions of the Rogers-Ramanujan Type. Duke Mathematical Journal, 32, 741-748.
[7] Connor, W.G. (1975) Partition Theorems Related to Some Identities of Rogers and Watson. Transactions of the American Mathematical Society, 214, 95-111.
[8] Hirschhorn, M.D. (1979) Some Partition Theorems of the Rogers-Ramanujan Type. Journal of Combinatorial Theory, Series A, 27, 33-37.
[9] Agarwal, A.K. and Andrews, G.E. (1986) Hook Differences and Lattice Paths. Journal of Statistical Planning and Inference, 14, 5-14.
[10] Subbarao, M.V. (1985) Some Rogers-Ramanujan Type Partition Theorems. Pacific Journal of Mathematics, 120, 431-435.
[11] Subbarao, M.V. and Agarwal, A.K. (1988) Further Theorems of Rogers Ramanujan Type. Canadian Mathematical Society, 31, 210-214.
[12] Agarwal, A.K. and Andrews, G.E. (1987) Rogers-Ramanujan Identities for Partitions with “N-Copies of N”. Journal of Combinatorial Theory, Series A, 45, 40-49.
[13] Agarwal, A.K. (1988) Rogers-Ramanujan Identities for n-Colour Partitions. Journal of Number Theory, 28, 299-305.
[14] Agarwal, A.K. (1989) New Combinatorial Interpretations of Two Analytic Identities. Proceedings of the AMS— American Mathematical Society, 107, 561-567.
[15] Agarwal, A.K. (1991) q-Functional Equations and Some Partition Identities, Combinatorics and Theoretical Computer Science (Washington, DC, 1989). Discrete Applied Mathematics, 34, 17-26.
[16] Agarwal, A.K. (1996) New Classes of Infinite 3-Way partition Identities. ARS Combi-natoria, 44, 33-54.
[17] Goyal, M. and Agarwal, A.K. Further Rogers-Ramanujan Identities for n-Colour Partitions, Utilitas Mathematica. (to appear).
[18] Agarwal, A.K. and Bressoud, D.M. (1989) Lattice Paths and Multiple Basic Hypergeometric Series. Pacific Journal of Mathematics, 136, 209-228.
[19] Slater, L.J. (1952) Further Identities of the Rogers-Ramanujan Type. Proceedings London Mathematical Society, 54, 147-167.

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