5 - Hypergeometric and Basic Hypergeometric Series and Integrals Associated with Root Systems

Summary

This chapter provides an overview of some of the main results from the theories of hypergeometric and basic hypergeometric series and integrals associated with root systems. In particular, a number of summations, transformations, and explicit evaluations for such multiple series and integrals is listed. The focus is on those results that do not directly extend to the elliptic level. The featured results include multivariate versions of the terminating q-binomial theorem, the q-Pfaff-Saalschütz summation, the Jackson summation, some multilateral summations including multivariate versions of Dougall's 2H2 summation, Ramanujan's 1psi1 summation, Bailey's 6psi6 summation, multivariate Watson and Bailey transformations, dimension changing transformations, and multidimensional generalizations of the Askey-Wilson integral evaluation. A survey on the theory of basic hypergeometric series with Macdonald polynomial argument is provided as well.