Fractional Symmetry and Comparison Structures of the Lerch Transcendent

Abstract

The Lerch transcendent is a three-parameter special function containing the Riemann zeta, Hurwitz zeta, Dirichlet eta, and polylogarithm functions as distinguished specializations. This study develops a defect-based framework for comparing fractional differentiation with parameter transformations and authentic functional covariances of Lerch-type functions. The fractional operator is fixed throughout as the right Weyl–Marchaud derivative in the spectral variable. First, this study separates arbitrary comparison pullbacks from genuine weighted covariance operators and establish explicit conditions under which the Lerch series may be differentiated termwise, including an integrable tail estimate that justifies interchange with the unbounded Weyl integral in the regimes used in the paper. Then, the full commutator of the fractional derivative with a weighted composition operator is analyzed, including the pullback contribution. Beyond the general formalism, the Hurwitz zeta duplication formula is studied as a concrete non-elementary covariance. Its fractional symmetry defect is obtained both as a convergent Marchaud integral and as an explicit Dirichlet series. For 02, and real s>1, this defect is strictly positive, showing that the classical duplication covariance is not preserved by the chosen fractional operator in that region. The geometrically forced fixed-point line used in the comparison examples and the numerical experiments is retained as a calibration benchmark rather than as evidence for a shifted critical strip. A concrete spectral-zeta realization is also provided for the affine-spectrum operator whose spectral zeta function is the Hurwitz zeta function.