An Invitation to the Rogers-Ramanujan Identities

This one-of-a-kind text, best suited for graduate level students and above, focuses exclusively on the Rogers-Ramanujan identities and their history. These two identities from number theory involve both infinite series and infinite products. The identities were independently discovered by Leonard James Rogers (1894 with proof), Srinivasa Ramanujan (before 1913 without proof), and Issai Schur (1917 with proof). The identities are relevant to the study of integer partitions, Lie algebras, statistical mechanics, computer science, and several other areas. Sills (Georgia Southern Univ.) begins with a review of partition theory and hypergeometric series. In the next two chapters, he moves on to prove the Rogers-Ramanujan identities and to explain their combinatorial aspects, as well as related identities and extensions. The final two chapters treat applications including continued fractions and knot theory. One appendix lists 236 related identities. A second appendix enhances the book's historical utility by providing transcriptions of letters between key researchers from 1943 to 1961. The book also includes more than 60 enlightening exercises. -D. P. Turner, Faulkner University, CHOICE Reviews In recent years -series have arisen in knot theory, and Sills discusses a few -series that arise in the study of certain simple fundamental knots. While there have been systematic discussions of many of the topics mentioned above in various books, monographs, and survey articles, Sills's book is the first comprehensive discussion of R-R type identities in all their forms, describing the state of the art. Since the subject is so vast, he does not provide proofs for most of the identities discussed, but he provides an interesting and illuminating historical context for each topic, gives good motivation, and describes the key ideas underlying the proofs. He also provides a substantial number of references that will lead both the student and the expert to some of the most important sources in the field. [. . .] In summary, this is a comprehensive and easily readable treatment of R-R type identities of appeal to both the expert and the potential entrant to the field. It will be a fine addition to both libraries and your personal book collection. -Krishnaswami Alladi, AMA Reviews