Hilfer–Taylor expansions and fractional Appell-type sequences

We develop a Hilfer-adapted Taylor-type framework that is compatible with the natural initial trace of the Hilfer fractional derivative. For $0<\alpha <1$ and $\beta \in [0,1]$, we introduce a shifted $(\alpha ,\beta )$-fractional power series (FPS) with $\delta =\alpha (1-\beta )+\beta -1$ and define Hilfer–Taylor coefficients via the regularized trace ${\mathcal{T}}_n(f)=(I^{(1-\beta )(1-\alpha )}(D^{\alpha ,\beta }{)}^{n}f)(0+)$. This yields an explicit coefficient formula and a Taylor-type expansion in the normalized basis $t^{n\alpha +\delta }/\mathrm{\Gamma }(n\alpha +\delta +1)$. Using the associated Mittag–Leffler eigenfunction kernel $G_{\alpha ,\delta }(t,x)=x^{\delta }{E}_{\alpha ,\delta +1}(t^{\alpha }{x}^{\alpha })$, we define fractional Appell-type sequences through a Hilfer-adapted generating identity and establish their main operational properties, including a lowering relation under $D_x^{\alpha ,\beta }$. As an application, we introduce Bernoulli-type objects and derive a convolution recurrence for the corresponding fractional Bernoulli numbers, recovering the classical case when $(\alpha ,\beta )=(1,1)$.