A Continuous-Order Integral Operator for Maclaurin-type Reconstruction

I introduce a continuous-order integral analog of the Maclaurin expansion that reconstructs analytic functions from fractional derivative data. This operator replaces the discrete sum of integer-order derivatives in the classical Maclaurin expansion with an integral over fractional derivative orders, weighted by a Maclaurin-type kernel. I show that, under smoothness and decay assumptions, the discrepancy between this construction and the classical Maclaurin series is governed by the Euler–Maclaurin summation formula, and I define a corresponding correction series. Numerical experiments on a representative set of analytic functions show that the uncorrected operator reliably tracks the global structure of $f$, with a systematic, mostly constant offset and additional deviation near the origin. Adding the first three correction terms substantially reduces these discrepancies across the tested domains, lowering the mean absolute error by two orders of magnitude, from $10^{-1}$ to $10^{-3}$. For monomials, the order data collapse to a single atomic contribution; using Caputo-consistent boundary data, the operator reconstructs them exactly. Altogether, these results show that this continuous-order operator, together with its correction series, provides a coherent framework for extending the classical Maclaurin expansion to spectral representations in derivative order.

💡 Research Summary

The paper proposes a novel “continuous‑order integral operator” that generalizes the classical Maclaurin series by replacing the discrete sum over integer‑order derivatives with an integral over fractional‑order derivatives. The operator is defined as

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