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) = \bigl(I^{(1-\beta)(1-\alpha)}(D^{\alpha, \beta})^n f\bigr)(0+) $. This yields an explicit coefficient formula and a Taylor-type expansion in the normalized basis $ t^{n\alpha+\delta}/\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) $.
Citation: Erick Castillo, Stiven Díaz, Juan Hernández, William Ramírez. Hilfer–Taylor expansions and fractional Appell-type sequences[J]. AIMS Mathematics, 2026, 11(6): 16613-16634. doi: 10.3934/math.2026682
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) = \bigl(I^{(1-\beta)(1-\alpha)}(D^{\alpha, \beta})^n f\bigr)(0+) $. This yields an explicit coefficient formula and a Taylor-type expansion in the normalized basis $ t^{n\alpha+\delta}/\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) $.
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Erick Castillo, Stiven Díaz, Juan Hernández, William Ramírez. Hilfer–Taylor expansions and fractional Appell-type sequences[J]. AIMS Mathematics, 2026, 11(6): 16613-16634. doi: 10.3934/math.2026682
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