Singularities and analytic continuation of the Dunkl and the Jacobi-Cherednik intertwining operators and their duals (Q450972)

This paper is devoted to the study of the domains of singularities in \(\mathbb C\) (resp., in \(\mathbb C\times \mathbb C)\) of the operator valued functions \(k\mapsto V_k\) and \((k,k')\mapsto V_{k,k'}\), where \(V_k\) and \(V_{k,k'}\) are the operators which intertwine the derivative operator \(\frac{d}{dx}\) with, respectively, the Dunkl operator NEWLINE\[NEWLINET(k) f (x) = \frac{df}{dx} (x) + k \frac{F(x) - f(-x)}{x}NEWLINE\]NEWLINE and the Jacobi--Cherednik operator \(T^{(k,k')}f(x)=f'(x)+(\text{kcoth}(x)+k' \text{tanh}(x))(f(x)-f(-x))-(k+k')f(-x)\). 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 \(\mathbb C\) and \(\mathbb C\times \mathbb C\), respectively.