Chapter 3 - The hypergeometric shift operators

Abstract

This chapter discusses the hypergeometric shift operators. That shift operators should exist for higher rank root systems was first hinted at by Koornwinder who found a shift operator for R of type BC2. A systematic study of shift operators was made by Opdam in his thesis. The chapter provides simplified treatment of the existence of shift operators. In particular shift operators are W-invariant differential operators on Hreg which map C[P]w into itself, and hence they can also be viewed as elements of some Weyl algebra An. The mapping is injective, and a shift operator of order N is mapped onto a polynomial of degree N.