Geodesic
Mean-Periodic Functions Associated with the Jacobi-Dunkl Operator on R
Fractional calculus and applied analysis, Tome 9 (2006) no. 3, pp. 215-236.

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Using a convolution structure on the real line associated with the Jacobi-Dunkl differential-difference operator Λα,β given by: Λα,βf(x) = f'(x) + ((2α + 1) coth x + (2β + 1) tanh x) { ( f(x) − f(−x) ) / 2 }, α ≥ β ≥ −1/2 , we define mean-periodic functions associated with Λα,β. We characterize these functions as an expansion series intervening appropriate elementary functions expressed in terms of the derivatives of the eigenfunction of Λα,β. Next, we deal with the Pompeiu type problem and convolution equations for this operator.
Keywords: Jacobi-Dunkl Operator, Mean Periodic Function, Jacobi-Dunkl Expansion, Pompeiu Problem 
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Ben Salem, N.; Ould Ahmed Salem, A.; Selmi, B. Mean-Periodic Functions Associated with the Jacobi-Dunkl Operator on R. Fractional calculus and applied analysis, Tome 9 (2006) no. 3, pp. 215-236. http://geodesic.mathdoc.fr/item/FCAA_2006_9_3_a1/