@article{Awadalla2025, 
author = {Muath Awadalla},
title = {Fractional Hermite functions associated with the Atangana–Baleanu Caputo derivative power series solutions, Rodrigues representation, and orthogonality analysis},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {9},
pages = {20586-20605},
keywords = {Atangana–Baleanu Caputo derivative, fractional Hermite functions, Mittag-Leffler kernel, Rodrigues-type formula, orthogonality in fractional calculus},
url = {https://www.sciopen.com/article/10.3934/math.2025919},
doi = {10.3934/math.2025919},
abstract = {This article established a comprehensive analytical framework for fractional Hermite functions using the Atangana-Baleanu Caputo (ABC) derivative. We derived a convergent power series solution (radius        |    x      |    <  1 for    α  ∈  (  0  ,  1  )) with explicit recurrence relations for its coefficients. Even and odd fractional Hermite functions were constructed via novel termination conditions, and a generalized Rodrigues-type formula was presented. A central result was the proof of orthogonality with respect to the weight function        W          α        (  x  )  =      e          −              x                  2                          E          α            (    −      α          1      −      α            |    x            |              2              /            α            )  , accompanied by the derivation of exact normalization constants        Λ          n        (  α  ). Numerical validation confirmed theoretical predictions, with errors    <  0.5  %  . The functions        H          n      ,      α              A      B      C        (  x  ) preserved key classical properties while exhibiting distinct fractional behavior, such as cusp-like formation at the origin. Quantitative analysis demonstrated convergence to classical Hermite polynomials as    α  →      1          −      , with root errors    <  1  % for    α  =  0.95. This work extends Hermite theory into the fractional domain, providing essential tools for modeling systems with memory and non-local interactions.}
}