Latest revision as of 09:46, 28 July 2025

Abstract

 T his paper develops a rigorous framework for fractional Hermite func tions using Caputo derivatives. We establish three fundamental contribu tions. First, we derive a complete power series solution to the fractional Hermite equation expressed through the basis {xkα}∞ k=0 , proving absolute convergence for all x ∈ C through careful asymptotic analysis of the coefficients. Second, we introduce and characterize both even and odd fractional Hermite functions H(α) n (x), deriving their explicit representa tions, recurrence relations, and a fractional Rodrigues-type formula that generalizes the classical case. Third, we demonstrate their orthogonality properties under the weight function wα(x) = |x|α−1e−|x|2α, obtaining exact normalization constants. The theoretical results are supported by comprehensive graphical analysis showing the systematic deformation of classical Hermite polynomial features as α varies. These developments provide new tools for fractional spectral methods and advance the under standing of orthogonal function systems in fractional calculus.

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