Computer algebra and identities of the Rogers-Ramanujan type.

摘要

In his last letter to Hardy, Ramanujan gave a list of functions, which he called "Mock theta functions," and just a few relations among them without any formal proof. His main object in this study was to discover new classes of functions enjoying many of the important properties of the classical theta functions. Each of Ramanujan's mock theta functions is closely related to a q-series expansion of a classical theta function. For example, the celebrated Rogers-Ramanujan identities have, according to Ramanujan, 10 mock theta functions arising from them.; Around 1950, L. J. Slater expanded classical work on Rogers-Ramanujan-type series to provide a list of more than 100 such identities. The central theme of this thesis is the exploration of q-series, like those of Slater, with the object of extending Ramanujan's study of mock theta functions and related mathematical objects.; In 1985, Andrews introduced a general method for extending Rogers-Ramanujan-type series to two variables so that Ramanujan's mock theta functions arise as specializations. We shall consider how extensively this approach provides interesting generalizations of Slater's identities.; In dealing with these ideas one can also provide an alternative proof for Slater's identities. In Chapters 2 and 6 we show how to do this for most of the identities in Slater.; Chapter 1 has the necessary notations, some well-known results and a new q-analog for the trinomial coefficients.; In Chapter 3 we look at the two-variable functions f(q, t) for values of t other than 1. This is where the possible mock theta functions applications are explored.; Chapter 4 has results in partitions that uses some of Slater's identities.; Chapter 5 has a result connecting single and double sums that allows us to simplify the proof, given by Andrews, for one of the Luztig-Macdonald-Wall conjectures.; The results in Chapters 3, 5 and particularly 6 would not be possible without the help of a symbolic algebra package. We have used "MACSYMA" to get this done.