Rogers - Ramanujan Type Identities and Split (n+t) - Color Partitions

Abstract

The present dissertation contains a detail study of investigations carried out by various authors and by us on existence of combinatorial interpretations of certain q – series. The whole work is divided into four chapters. Chapter 1 is introductory which includes elementary definitions and results which will be required for the later chapters. It also includes a theorem in classical ordinary partitions proved by Agarwal [“On a generalized partition theorem”, J. Indian Math. Soc.50(1986), pp. 185-190.]. In Chapter 2, the main theorem of Chapter 1 is extended to colored partitions proved by Agarwal and Rana [“Two new combinatorial interpretations of a fifth order mock theta function”, The Ind. Math. Soc., Special Centenary Volume, 1907-2007,(2008), 11 - 24]. This chapter is further devoted to the study of (n+t) – colored partitions. In Chapter 3, a new class of partitions which has been defined by Agarwal and Sood [“Split (n+t) – color partitions and Gordon – McIntosh eight order mock theta functions”, Electronic J. Comb.No. 2, 21(2014), p2.46], called split (n+t) – color partitions. The purpose of this new class of partitions is to interpret combinatorially some q - series which have not been interpreted by previous tools. This class also extends the color partitions. Finally in Chapter 4, we give some new combinatorial interpretations using split (n+t) – color partitions introduced in Chapter 3. We had given four generalized theorems which further give rise to four Rogers – Ramanujan Type Identities as a particular cases. Towards the end, references of various publications cited in the present dissertation have been reported.