Parametric Integral Symmetry and the Analytical Derivation of the K-Legendre Duplication Identity

This research presents an alternative analytical path to the k-generalized Legendre duplication formula, a pivotal identity in the study of k-calculus[1][2].Movingaway from traditional proofs based on the Bohr-Mollerup theorem, we employ aparametric integral transformation technique[2][3]. By analyzing the symmetry ofthe k-Beta function kernel, we demonstrate a derivation that is both intuitive andcomputationally verifiable[4]. This approach is particularly relevant for modelingfractional-order systems in engineering[5], where the k-Gamma function serves as acore component of generalized Laplace transforms[6].