Yasushi KAJIHARA

In this talk, I will study some hypergeometric transformation and summation formulas for a Milne's class of multiple generalization of (ordinary, basic and elliptic) hypergeometric series by starting from the Cauchy's reproducing kernel for symmetric function and the Cauchy determinant for the Weierstrass sigma function. First, I will derive a multiple Euler transformation for hypergeometric series by using symmetries of Cauchy kernel and Macdonald's q-difference operator and by a method of multiple principal specialization. As an application of it, a number of multiple hypergeometric transformation and summation formulas including very well-poised hypergeometric series will be given. Among them, in particular, I will propose the duality transformation formula and its balanced case. By using these formulas, two types of multiple Bailey's

transformation formula will be derived and will be given further transformations and sums including already known ones.