RAMANUJAN’S SPT-CRANK FOR MARKED OVERPARTITIONS

Nil Ratan Bhattacharjee; Sabuj Das

doi:10.29121/granthaalayah.v3.i8.2015.2958

Authors

Nil Ratan Bhattacharjee Professor of Mathematics, University of Chittagong, Bangladesh
Sabuj Das Senior Lecturer, Department of Mathematics, Raozan University College, Bangladesh

DOI:

https://doi.org/10.29121/granthaalayah.v3.i8.2015.2958

Keywords:

Components, Congruent, Crank, Overpartitions, Overlined, Weight

Abstract [English]

In 1916, Ramanujan’s showed the spt-crank for marked overpartitions. The corresponding special functions , and are found in Ramanujan’s notebooks, part 111.

In 2009, Bingmann, Lovejoy and Osburn defined the generating functions for ,

and . In 2012, Andrews, Garvan, and Liang defined the in terms of partition pairs. In this article the number of smallest parts in the overpartitions of n with smallest part not overlined, not overlined and odd, not overlined and even are discussed, and the vector partitions and - partitions with 4 components, each a partition with certain restrictions are also discussed. The generating functions , , , , are shown with the corresponding results in terms of modulo 3, where the generating functions , are collected from Ramanujan’s notebooks, part 111. This paper shows how to prove the Theorem 1 in terms of ,Theorem 2 in terms of and Theorem 3 in terms of respectively with the numerical examples, and shows how to prove the Theorems 4,5 and 6 with the help of in terms of partition pairs. In 2014, Garvan and Jennings-Shaffer are able to defined the for marked overpartitions. This paper also shows another results with the help of 6 -partition pairs of 3, help of 20 -partition pairs of 5 and help of 15 -partition pairs of 8 respectively.

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References

G.E. Andrews, F. Dyson, and R. Rhoades. On the distribution of the spt-crank. Mathematics, 1(3),76–88, 2013. DOI: https://doi.org/10.3390/math1030076

G.E. Andrews, F.G. Garvan, and J. Liang. Combinatorial interpretations of congruences for the spt-function. RamanujanJ.29(1-3):321–338, 2012. DOI: https://doi.org/10.1007/s11139-012-9369-7

A. Berkovich and F.G. Garvan. K.Saito’s conjecture for nonnegative eta products and analogous results for other infinite products. J. Number Theory, 128(6):1731–1748, 2008. DOI: https://doi.org/10.1016/j.jnt.2007.02.002

K. Bringmann, J. Lovejoy, and R. Osburn. Rank and crank moments for overpartitions J. Number Theory, 129(7):1758–1772, 2009. DOI: https://doi.org/10.1016/j.jnt.2008.10.017

K. Bringmann, J. Lovejoy, and R. Osburn. Automorphic properties of generating

functions for generalized rank moments and Durfee symbols. Int. Math. Res. Not. IMRN,

(2):238–260, 2010.

] B.C. Berndt, Ramanujan’s notebooks. Part 111. Springer-Verlag, New York, 1991. DOI: https://doi.org/10.1007/978-1-4612-0965-2

W.Y.C. Chen, K.Q. Ji, and W.J.T. Zang. The spt-crank for ordinary Partitions. ArXiv e-prints, Aug.2013.

F.G. Garvan and Chris Jennings-Shaffer, The spt-crank for overpartitionsar

Xiv:1311.3680v2[Math.NT],23 Mar 2014.

J. Lovejoy and R. Osburn. M2-rank differences for partitions without repeated odd parts. J. Theor.Nombres Bordeaux, 21(2):313-334, 2009. DOI: https://doi.org/10.5802/jtnb.673

Chris Jenning-Shaffer, Another proof of two modulo 3 congruences and another spt crank for thenumber of smallest parts in overpartitions with even smallest part, arXiv:1406.5458v1 [Math.NT] 20 Jun 2014.