A Generalized Hypergeometric Function¶Satisfying Four Analytic Difference Equations¶of Askey--Wilson Type

The hypergeometric function 2 F 1 can be written in terms of a contour integral involving gamma functions. We generalize this (Barnes) representation by using a certain generalized gamma function as a building block. In this way we obtain a new 2 F 1-generalization with various symmetry features. We determine the analyticity properties of the R-function in all of its eight arguments, and show that it is a joint eigenfunction of four distinct Askey–Wilson type difference operators, two acting on v and two on . The Askey–Wilson polynomials can be obtained by a suitable discretization of v or .