### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 030, 16 pages      arXiv:0903.1609      http://dx.doi.org/10.3842/SIGMA.2009.030
Contribution to the Special Issue on Dunkl Operators and Related Topics

### Nonlocal Operational Calculi for Dunkl Operators

Ivan H. Dimovski and Valentin Z. Hristov
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 8, 1113 Sofia, Bulgaria

Received October 15, 2008, in final form March 04, 2009; Published online March 09, 2009

Abstract
The one-dimensional Dunkl operator Dk with a non-negative parameter k, is considered under an arbitrary nonlocal boundary value condition. The right inverse operator of Dk, satisfying this condition is studied. An operational calculus of Mikusinski type is developed. In the frames of this operational calculi an extension of the Heaviside algorithm for solution of nonlocal Cauchy boundary value problems for Dunkl functional-differential equations P(Dk)u = f with a given polynomial P is proposed. The solution of these equations in mean-periodic functions reduces to such problems. Necessary and sufficient condition for existence of unique solution in mean-periodic functions is found.

Key words: Dunkl operator; right inverse operator; Dunkl-Appell polynomials; convolution; multiplier; multiplier fraction; Dunkl equation; nonlocal Cauchy problem; Heaviside algorithm; mean-periodic function.

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