Study of ordinary coloured and signed gollnitz-gordon identities

Abstract

In this thesis, we studied combinatorial interpretation of generalized $q$ - series given by Agarwal in 1986 (A.K. Agarwal, On a Generalized Partition Theorem, J. Indian Math. Soc. Vol. 50 (1986),$~$ pp.185-190) using ordinary partitions. In 2009 Agarwal and Rana (A.K. Agarwal and M. Rana, New Combinatorial Version of G\"{o}llnitz - Gordon identities, Utilitas Mathematica., Vol. 79(2009),$~$ pp.145-156) extended the interpretation given by Agarwal in 1986 using $n$ - coloured partitions.In some particular cases these 2 - way combinatorial identities are extended to a 3 - way combinatorial identities which gives combinatorial interpretations of G\"{o}llnitz-Gordon identities using ordinary and $n$ - colour partition discussed in Chapter 2. We have also explored the signed G\"{o}llnitz- Gordon identities which is due to Andrews .sills(G.E.Andrews, Euler's De Partition Numerorum,bull.Amer.Math.Soc. Vol 44 (2007),561-573) and bijection between ordinary and signed G\"{o}llnitz - Gordon identities due to Andrews V. Sills ( Andrews V. Sills , On the Ordinary and Signed G\"{o}llnitz - Gordon Partitions , 2007. ) discussed in Chapter 3. Chapter 1 is devoted to elementary study of partition Theory.