This paper introduces and systematically studies a new class of non-commutative algebras -- Weyl-type and Witt-type algebras -- generated by differential operators with exponential and generalized power function coefficients. We define the expolynomial ring $R_{ p,t,\mathcal{ A}} = \mathbb{ F}\bigl[ e^{ \pm x^{ p} e^{ t}},\; e^{ \mathcal{ A} x},\; x^{ \mathcal{ A}} \bigr]$ associated to an additive subgroup $\mathcal{ A} \subset \mathbb{ F}$, and investigate its Ore extension $A_{ p,t,\mathcal{ A}} = R_{ p,t,\mathcal{ A}}[\partial; \delta]$ (Weyl-type) and its derivation algebra $\mathfrak{ g}_{ p,t,\mathcal{ A}} = \operatorname{ Der}_{ \mathbb{ F}}(R_{ p,t,\mathcal{ A}})$ (Witt-type). Our main results establish: (1) the automorphism group of $R_{ p,t,\mathcal{ A}}$ is isomorphic to $(\mathbb{ F}^{ \times})^{ 2r+1} \rtimes \operatorname{ GL}(2r+1,\mathbb{ Z})$; (2) a Galois descent theorem showing that fixed-point subalgebras under finite Galois actions recover the original Weyl-type algebra; (3) the non-existence of finite-dimensional simple modules for $A_{ p,t,\mathcal{ A}}$; (4) the Zariski density of isomorphism classes in moduli spaces as transcendental parameters vary; (5) the stability of simplicity under generic quantum deformation; and (6) a complete representation-theoretic framework including the classification of irreducible weight modules, the construction of Harish--Chandra modules with BGG-type resolutions, and the structure of category $\mathcal{ O}$. These results unify and extend classical theories of Weyl algebras, Witt algebras, and generalized Weyl algebras, while opening new directions in deformation theory, non-commutative geometry, and the representation theory of infinite-dimensional algebras.
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