Singularities and analytic continuation of the Dunkl and the Jacobi-Cherednik intertwining operators and their duals

This paper is devoted to the study of the domains of singularities in C (resp. in C×C) of the operator valued functions k↦Vk and (k,k′)↦Vk,k′, where Vk and Vk,k′ are the operators which intertwine the derivative operator ddx with respectively the Dunkl operator T(k)f(x)=dfdx(x)+kf(x)−f(−x)x and the Jacobi-Cherednik operator T(k,k′)f(x)=f′(x)+(kcoth(x)+k′tanh(x))(f(x)−f(−x))−(k+k′)f(−x). We also determine the singularities of the inverses and the duals of these operators Vk and Vk,k′ by analytic methods and we show that some of them are entire functions whereas others are only meromorphic functions on C and C×C respectively.