Some Recent Developments in q - Series ( Extended

In this talk, I shall concentrate on q-series algorithms that have been implemented in computer algebra. This, of course, ignores a variety of other work (including some described in my other talk). However, I believe that computational techniques arising from advances in computer algebra make one of the most important and appealing topics in this subject. There are essentially four topics that will be considered here primarily because they have been important to my work in various ways. I hope the authors of other algorithmic methods will forgive me for my narrow approach. First we jointly consider the celebrated W − Z method [17] and Pfaff’s method [2], [15]. These methods rely on contiguous relations to prove identities. The work in [17], [18], [19] has been successfully implemented in MAPLE by D. Zeilberger (cf. [14]) in order to provide breath takingly simple proof of famous theorems in ordinary and basic hypergeometric series. It should be added that improvements and extensions of the WZ method form a very active area of research. Among those contributing are F. Chyzak [8], P. Paule and A. Riese [13] and K. Wegschaider [16]. Much greater detail can be gained from the wonderful book by M. Petkovs̆ek, H. Wilf and D. Zeilberger [14]. Let us consider a classic example from ordinary hypergeometric series to illustrate our viewpoint: