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In this paper we study the homogeneous Wick differential equation associated to the quantum white noise () Euler operatorΔ  E   g  ,  Q$\Delta _E^{g, Q}$    acting on generalized operators.Δ  E   g  ,  Q$\Delta _E^{g, Q}$    is defined as sum of the extension of the -Gross Laplacian and the -conservation operator. It is shown that the operatorΔ  E   g  ,  Q$\Delta _E^{g, Q}$    has a representation in terms of the -derivatives{   D  c  −   ,   D  c  +   :    c  ∈  N  }   $\{ D_c^ -, D_c^ +: \;c \in N\} $   . The poisson equation is worked out as a non homogeneous Wick differential equation associated toΔ  E   g  ,  Q$\Delta _E^{g, Q}$   .

Wick differential and Poisson equations associated to the 𝚀𝚆𝙽-Euler operator acting on generalized operators