Mittag-Leffler Weighted Orthogonal Functions for the ABC Fractional Operator: A Formal Self-Adjointness Construction

This work constructs an orthogonal function system on bounded intervals [0,R] associated with the Atangana–Baleanu–Caputo (ABC) fractional derivative for α∈(1/2,1). Starting from a fractional Laguerre-type equation involving the ABC operator, solutions are obtained via a generalized Frobenius method, yielding series representations with characteristic exponent α−1. Rather than postulating a weight function by analogy with classical or Caputo settings, the weight is derived directly from the requirement that the underlying fractional operator be formally self-adjoint on a suitable admissible domain. This operator-theoretic approach leads to the explicit Mittag–Leffler weight wα(x)=x−(2α−1)Eα(−xα), which intrinsically reflects the nonlocal memory structure of the ABC kernel. A similarity transformation removes the universal singular factor and produces regularized eigenfunctions that are continuous on [0,R] and orthogonal in the weighted L2 space. The weight identity and formal self-adjointness are rigorously verified through a right-Volterra uniqueness argument. Numerical experiments confirm orthogonality up to machine precision, demonstrate spectral convergence for a model ABC differential equation, and illustrate consistency with classical Laguerre polynomials in the limit α→1−. The resulting framework provides a self-consistent orthogonal system suitable for spectral approximations of problems governed by the ABC operator on bounded domains.