Rogers-Ramanujan Type Identities and Combinatorics

Abstract

In this thesis, we interpret several q–series and q–identities employing combinatorial tools of partitioning of integers, such as (n+t)–color partitions introduced by Agarwal and Andrews in 1987 (Agarwal, A. K. and Andrews, G. E. Rogers–Ramanujan identities for partitions with “N copies of N”. Journal of Combinatorial Theory, Series A, 45:40–49, 1987), lattice paths defined by Agarwal and Bressoud in 1989 (Agarwal, A. K. and Bressoud, D. Lattice paths and multiple basic hypergeometric series. Pacific Journal of Mathematics, 136:209–228, 1989) and F–partitions introduced by Andrews in 1984 (Andrews, G. E. Generalized Frobenius partitions. American Mathematical Society, 301, 1984). We have obtained four–way combinatorial indentities. Each four–way combinatorial identity gives us six new combinatorial identities in the usual sense and we get a total of eighteen new combinatorial identities. These new results are contained in Chapter 2 and Chapter 4. The results obtained are accepted for publication as per details given below: • Sareen, J. K. and Rana, M. Four–way combinatorial interpretations of some Rogers–Ramanujan type identities (Accepted). Ars Combinatoria, 2014 (SCI, Impact Factor 0.259). In Chapter 3 we interpret two tenth order mock theta functions combinatorially using (n + t)–color partitions and two mock theta functions generated by Gordon and McIntosh in 2000 (Gordon, B. and McIntosh, R. J. Some eighth order mock theta functions. Journal of the London Mathematical Society, 62:321–335, 2000) using signed partitions and ordinary partitions. We have further extended the combinatoix Abstract rial interpretation of one of the tenth order mock theta function using F–partitions explicitly given in Chapter 4. The results obtained are accepted/published as per details given below: • Sareen, J. K. and Rana, M. Combinatorics of tenth order mock theta functions (Accepted). Proceedings of the Indian Academy of Sciences–Mathematical Sciences, 2016 (SCI, Impact Factor 0.240). • Rana, M. and Sareen, J. K. On combinatorial extensions of some mock theta functions using signed partitions. Advances in Theoretical and Applied Mathematics, 10(1):15–25, 2015. Chapter 5 is based on combinatorial interpretations of generalized q–series and split (n + t)–color partitions. Each generalized q–series given in this chapter is in conjunction with a Rogers–Ramanujan type identity for a particular value of the parameter. The results obtained in this chapter are accepted for publication as per details given below: • Rana, M., Sareen, J. K. and Chawla, D. On generalized q–series and split (n + t)–color partitions (Accepted). Utilitas Mathematica, 2015 (SCI, Impact Factor 0.354). Further in Chapter 6, the results of Chapter 5 are extended and analogues to the bijections between (n + t)–color partitions and F–partitions, new bijections between split (n + t)–color partitions and 2–color F–partitions are established for the generalized q–series and hence for Rogers–Ramanujan type identities. Also the similar bijections are established for two Gordon–McIntosh mock theta functions. The results obtained in this chapter are communicated for publication as per details given below: • Rana, M. and Sareen, J. K. Split (n + t)–color partitions and 2–color F– partitions (Communicated).