A Unified Gegenbauer Framework for Arbitrary-Order Riemann-Liouville Fractional Integrals with Tunable Endpoint Regularity

We show that the GBFA method of Elgindy (2025), originally developed for 0 < α 0 without altering its algorithmic structure. Elgindy's transformation τ = t(1 − y1/α) remains valid for all α > 0 and preserves the numerical framework, ensuring that interpolation, quadrature, and error analysis carry over unchanged. For α > 1, the mapping induces only Hölder regularity at y = 0, which a ects quadrature accuracy. We quantify this e ect and show that the interpolation error retains its original convergence properties. To restore higher-order endpoint smoothness, we introduce a ϕ(α)-generalized transformation that enforces Cr regularity for any prescribed r ≥ 0, accelerating quadrature convergence while preserving the GBFA structure. Numerical experiments con rm high accuracy and robustness across all α > 0, demonstrating that the uni ed GBFA formulation provides an e cient, non-adaptive, xed-node approach for arbitrary-order RLFIs.