Functions of hypergeometric type

Abstract

This thesis deals with a method of expressing, as infinite products, some special limiting cases of a general transformation (2,4) between basic hypergeometric series. A short historical introduction (1.2), is followed by some preliminary theorems on the transformation of infinite series (2.1)--(2.3), a proof of the basic transformation, and a discussion of the work of L.J. Rogers (3.1)--(3.2). Rogers expressed a number of these limiting cases of hypergeometric functions in terms of infinite products by using trigonometrical identities, and, in particular, a sketch is given of his method of deducing the group of series called by him, A-series.

A new method is then given (4.1)--(4.6) of deducing these A-series, using basic bilateral series. By employing W.N. Bailey's summation theorem for the bilateral series, all the transformations given by Rogers, are deduced, together with a number of new transformations, (4.7). Limiting cases of these special transformations are then considered (5.1.)--(5.3) and lead to the deduction of a large number of special identities of the Rogers-Hamanujan type, (5.4)--(6.3). Of these about forty involving products of the types [equations] are believed to be new.

Two proofs are then given of the summation theorem (7.1)--(7.2), and a generalisation (7.4) of the original basic transformation, some equivalent product theorems (5.3), (7.3) and (7.5) are also considered, end the thesis concludes with an appendix containing a list of one hundred and thirty identities, which have been deduced in the body of the thesis.

Three papers, which together form a shortened version of this thesis have been accepted for publication by the London Mathematical Society, under the titles "A New Proof of Rogers's Transformations of Infinite series". Proceedings, L.M.S. Vol. 53. pp 460--475. "Further identities of the Rogers-Ramanujan Type" Proceedings L.M.S. (In course of publication) "A Note on Equivalent Product Theorems". Journal, L.M.S. (In course of publication).