Dunkl derivative from moment differentiation
Edmundo J. Huertas, Alberto Lastra, Judit Minguez Ceniceros
TL;DR
The paper unifies Dunkl operator theory with the moment-derivative framework by identifying the Dunkl operator as the moment derivative corresponding to the Dunkl factorial sequence $\gamma_{p,\alpha}$. It develops linear moment-differential systems, derives explicit solutions via the Dunkl exponential $E_{\alpha}$, and analyzes their asymptotics under the strongly regular sequence $\gamma_{p,\alpha}$. It further extends the theory to generalized translations through $m$-translation and introduces an $m$-even translation to isolate parity effects, with spectral actions on $E_m^{\xi}$ clarifying operator structure. The work creates a mutualistic bridge between moment-differentiable and Dunkl theories, enabling generalized summability and functional-equation analyses in a broader translational framework.
Abstract
The work analyzes the theory of Dunkl operator as a moment differential operator. This last operator generalizes the first one whenever the sequence of moments satisfies appropriate classical properties, classically considered in the general theory of ultraholomorphic and ultradifferentiable classes of functions. In this sense, the theory of Dunkl operator is then generalized. On the other hand, some features developed in Dunkl theory, such as Dunkl translation, have not been considered in the theory of moment differential equations yet, which leads to a common mutualism involving both theories.