Generalization of the Dunkl operator on the space of entire functions (Q954193)

Let \(H(\mathbb C)\) be the space of all entire functions. The classical one-dimensional Dunkl operator on \(H(\mathbb C)\) is defined as \[ \Lambda_\alpha f(z)= f'(z)+(\alpha+\frac12)\,\frac{f(z)-f(-z)}z, \quad \alpha>-\frac12, \quad f\in H(\mathbb C). \] In the paper under review the authors study a generalization of the Dunkl operator. The generalized Dunkl operator has the form \[ L(f)(z)=\sum_{n=0}^q A_n z^n f^{(n+1)}(z)+ \sum_{n=0}^p B_n z^n f^{(n+1)}(-z) +C\,\frac{f(z)-f(-z)}z, \] where the coefficients \(A_n, B_n\) and \(C\in\mathbb C\) satisfy the conditions (i) \(C\neq 0\) and (ii) \(A_0\neq 0\) or \(B_0\neq 0\). Using the generalized Dunkl operator, the authors introduce generalized translation and generalized convolution operators and derive several assertions that generalize the corresponding results concerning the Dunkl operator.