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Basic hypergeometric series and applications. With a foreword by George E. Andrews. (English) Zbl 0647.05004
Mathematical Surveys and Monographs, 27. Providence, RI: American Mathematical Society (AMS). xiii, 124 p. (1988).

One of the problems of working with basic hypergeometric series is the scarcity of good reference books. A measure of this scarcity is that although it has taken roughly thirty years for this book to see print, it still meets a very real need for a good introduction to this subject.

The book moves in a natural progression from techniques of deriving identities for basic hypergeometric series, through applications to partition theory and especially the Ramanujan partition congruences, into mock theta functions, and ultimately to results in modular equations. The book is a treasure house of new and little known identities and relationships. Chapter notes by George Andrews tie Fine’s identities to the current literature.

MSC:
 33D15 Basic hypergeometric functions of one variable, ${}_{r}{\varphi }_{s}$ 33-02 Research monographs (special functions) 05A15 Exact enumeration problems, generating functions 11P82 Analytic theory of partitions 11F37 Forms of half-integer weight, etc.