Generating functions and formal dilation structures for a fractional Hermite family

We develop a generating function framework for a fractional Hermite family $\{H_n^{(\alpha )}(x)\}$ defined through an explicit gamma function representation, within the setting of formal power series. The gamma normalized generating function ${\mathcal{G}}_{\alpha }(x,t)=\sum _{n=0}^{\mathrm{\infty }}{H}_n^{(\alpha )}(x)\frac{t^n}{\mathrm{\Gamma }(\alpha n+1)}.$ encodes the entire family in a single algebraic object. Working in the formal power series ring $\mathbb{C}[x^{\alpha }][[t]]$, we establish several structural properties of ${\mathcal{G}}_{\alpha }(x,t)$. We derive coefficient extraction and reconstruction formulas, obtain explicit double-sum representations, and establish an even–odd decomposition of the generating series. These results show that the generating function completely determines the underlying fractional Hermite family. We further introduce the Euler operator ${\mathcal{E}}_t=t{\mathrm{\partial }}_t$ and develop a formal functional calculus in the index variable. This leads to the dilation identity $\exp \left(s{\mathcal{E}}_t\right){\mathcal{G}}_{\alpha }(x,t)={\mathcal{G}}_{\alpha }(x,e^{s}t),$ together with the associated semigroup and infinitesimal identities. As additional algebraic consequences, we establish a homogeneous decomposition of the formal power series space, characterize the corresponding homogeneous components as eigenspaces of the Euler operator, construct the associated projection operators, derive commutation relations, and prove the invariance of the even and odd subspaces under the Euler action. All results are obtained within a purely formal coefficientwise framework and require no additional analytic assumptions. The resulting theory provides a consistent algebraic setting for studying this fractional Hermite family through generating function methods and suggests possible extensions to other classes of fractional special functions.