Rogers-Ramanujan Type Identities And False Theta Function Formulae

By means of the bilateral Bailey lemma and transformation formulae from unilateral series to bilateral ones,this thesis will systematically investigate Rogers-Ramanujan type identities and false theta function formulae.Most of the known identities of both kinds are reviewed and numerous new formulae are derived.The content is summarized as follows:1.The bilateral Bailey lemma is established by combining series rearrangement with the q-Pfaff-Saalsch(u|¨)tz formula,which leads consequently to a general transformation formula expressing unilateral series in terms of bilateral one.By utilizing two fundamental bilateral q-series identities due to Ramanujan and Bailey as well as reduced forms of the q-Dougall sum,thirty-five transformation formulae between unilateral and bilateral series are proved.2.By specializing the parameters appeared in the transformation formulae,numerous identities of the Rogers-Ramanujan type can be obtained.Among them,two hundred selected ones are displayed in details,including most of Slater's collection of 130 idenities and several new identities.Considering the importance of Slater's monumental work,a comparison is made for the identities between those treated by Slater and the present work.3.Application of the bilateral Bailey lemma along another direction is systematically examined,which results in a large number of false theta function formulae,containing most of the known ones discovered mainly by Rogers(1917) and Ramanujan(＜1920) in his "Lost Notebook".Four hundred false theta function formulae are exhibited with most of them having not appeared previously.