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 $W_{\alpha }(x)=e^{-x^2}{E}_{\alpha }(-\frac{\alpha }{1-\alpha }|x{|}^{2/\alpha })$, accompanied by the derivation of exact normalization constants ${\mathrm{\Lambda }}_n(\alpha )$. Numerical validation confirmed theoretical predictions, with errors $<0.5\mathrm{\%}.$ The functions $H_{n,\alpha }^{ABC}(x)$ 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 $<1\mathrm{\%}$ 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.