Fractional Hermite functions associated with the Atangana–Baleanu Caputo derivative power series solutions, Rodrigues representation, and orthogonality analysis

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 $ \alpha\in(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_{\alpha}(x) = e^{-x^{2}}E_{\alpha}\bigl(-\frac{\alpha}{1-\alpha}|x|^{2/\alpha}\bigr) $, accompanied by the derivation of exact normalization constants $ \Lambda_{n}(\alpha) $. Numerical validation confirmed theoretical predictions, with errors $ < 0.5\%. $ The functions $ H^{ABC}_{n, \alpha}(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 $ \alpha\to 1^{-} $, with root errors $ < 1\% $ for $ \alpha = 0.95 $. This work extends Hermite theory into the fractional domain, providing essential tools for modeling systems with memory and non-local interactions.