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 ↦ V k and ( k , k ′ ) ↦ V k , k ′ , where V k and V k , k ′ are the operators which intertwine the derivative operator d d x with respectively the Dunkl operator T ( k ) f ( x ) = d f d x ( x ) + k f ( x ) − f ( − x ) x and the Jacobi–Cherednik operator T ( k , k ′ ) f ( x ) = f ′ ( x ) + ( k coth ( 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 V k and V k , 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.