This research presents an alternative analytical path to the k-generalized Legendre duplication formula, a pivotal identity in the study of k-calculus12.Movingaway from traditional proofs based on the Bohr-Mollerup theorem, we employ aparametric integral transformation technique23. By analyzing the symmetry ofthe k-Beta function kernel, we demonstrate a derivation that is both intuitive andcomputationally verifiable4. This approach is particularly relevant for modelingfractional-order systems in engineering5, where the k-Gamma function serves as acore component of generalized Laplace transforms6.
Abhishek TV (Sat,) studied this question.