Computer-assisted explorations of Rogers-Ramanujan type identities

In 1894, L. J. Rogers of the University of Leeds discovered a pair of series-product identities that later became known as the "Rogers-Ramanujan Identities." During the first half of the twentieth century, a number of mathematicians, including Rogers, F. H. Jackson, and W. N. Bailey, discovered a number of identities which resembled the Rogers-Ramanujan identities in form. In her Ph.D. thesis in 1950, Lucy J. Slater (a student of Bailey) produced a list of 130 identities of Rogers-Ramanujan type. In Chapter 1, the relevant mathematical background and some history is discussed.; In 1986, George Andrews published an algorithm (referred to herein as "the method of nonhomogeneous first order q-difference equations,") by which polynomial generalizations of Rogers-Ramanujan identities could be obtained. In this present work, I refined the Andrews algorithm and implemented it in the computer algebra system Maple V in a package I have entitled "RRtools." As such, I was able to produce polynomial identities which converge to each of the series-product identities on Slater's list. Chapter 2 contains a description of the algorithm. Chapter 3 contains the list of polynomial identities.; The issue of proving identities conjectured with the aid of the RRtools package is taken up in Chapter 4. Particular emphasis is given to the automated proof theory of Wilf and Zeilberger, as well as the recpf package I wrote to aid in the production of proofs by recurrence.; Reciprocal dual identities are addressed in Chapter 5.; By relaxing the condition of requiring first order q-difference equations, I was able to find alternate polynomial generalizations for several Rogers-Ramanujan type identities. Chapter 6 contains a discussion of the relaxed version of the q-difference equation method and the new polynomial identities.; Chapters 7 and 8 are users guides for the RRtools and recpf packages respectively. Appendix 1 is an annoted version of Slater's original list of Rogers-Ramanujan type identities. Appendix 2 is a CD ROM containing Maple worksheets with recurrence style proofs of the identities from Chapter 3*; *This dissertation includes a CD that is compound (contains both a paper copy and a CD as part of the dissertation). The CD requires the following application: Maple V.