@article{Castillo2026, 
author = {Erick Castillo and Stiven Díaz and Juan Hernández and William Ramírez},
title = {Hilfer–Taylor expansions and fractional Appell-type sequences},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {6},
pages = {16613-16634},
keywords = {Hilfer fractional derivative, fractional Taylor formula, Mittag–Leffler function, fractional Appell-type sequences, Bernoulli sequences},
url = {https://www.sciopen.com/article/10.3934/math.2026682},
doi = {10.3934/math.2026682},
abstract = {We develop a Hilfer-adapted Taylor-type framework that is compatible with the natural initial trace of the Hilfer fractional derivative. For    0  <  α  <  1 and    β  ∈  [  0  ,  1  ], we introduce a shifted    (  α  ,  β  )-fractional power series (FPS) with    δ  =  α  (  1  −  β  )  +  β  −  1 and define Hilfer–Taylor coefficients via the regularized trace              T        n    (  f  )  =      (        I          (      1      −      β      )      (      1      −      α      )        (      D          α      ,      β            )    n    f      )    (  0  +  ). This yields an explicit coefficient formula and a Taylor-type expansion in the normalized basis        t          n      α      +      δ            /    Γ  (  n  α  +  δ  +  1  ). Using the associated Mittag–Leffler eigenfunction kernel        G          α      ,      δ        (  t  ,  x  )  =      x    δ        E          α      ,      δ      +      1        (      t    α        x    α    ), 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          α      ,      β      . As an application, we introduce Bernoulli-type objects and derive a convolution recurrence for the corresponding fractional Bernoulli numbers, recovering the classical case when    (  α  ,  β  )  =  (  1  ,  1  ).}
}