Heckman-opdam Hypergeometric Functions and Their Specializations

is completely integrable and hence L(k) is in a commuting system of differential operators with n algebraically independent operators. Then we have the following fundamental result (cf. [1]). Theorem [Heckman, Opdam]. When kα are generic, the function F (λ, k;x) has an analytic extension on R and defines a unique simultaneous eigenfunction of the commuting system of differential operators with the eigenvalue parametrized by λ so normalized that the eigenfunction takes the value 1 at the origin. Heckman-Opdam hypergeometric system of differential equations characterizing F (λ, k;x), which will be denoted by (HO), is a multi-variable analogue of a “rigid local system” among completely integrable quantum systems and we study three types of specializations of the system and the function F (λ, k;x) as follows.