This paper investigates a broad class of non-Gaussian measures, \( \mu _\Psi \) , associated with a family of generalized Wright functions, \(_m\Psi _q\) . First, we study these measures in Euclidean spaces \(\mathbb {R}^d\) , then define them in an abstract nuclear triple \(\mathcal {N}\subset \mathcal {H}\subset \mathcal {N}'\) . We study analyticity, invariance properties, and ergodicity under a particular group of automorphisms. Then we show the existence of an Appell system which allows the extension of the non-Gaussian Hilbert space \(L^2(\mu _\Psi )\) to the nuclear triple consisting of test functions and distributions spaces, \((\mathcal {N})^{1}\subset L^2(\mu _\Psi )\subset (\mathcal {N})_{\mu _\Psi }^{-1}\) . Furthermore, thanks to the definition of two transformations, \(S_{\mu _{\Psi }}\) and \(T_{\mu _{\Psi }}\) , we study Donsker’s delta as an element within \((\mathcal {N})_{\mu _\Psi }^{-1}\) applying the integral equations fulfilled by \(_m\Psi _q\) .
