A note on the generalized ML function as the kernel of the extended wright function

This paper introduces a unified generalization of the Gamma and Beta functions by employing the generalized Mittag-Leffler function as the kernel. The resulting extended Wright function incorporates a newly proposed generalized Beta function that enhances analytical flexibility in modeling fractional systems. Several integral transforms are established along with Riemann–Liouville and Hilfer fractional derivative properties. These results generalize many classical identities and offer potential applications in fractional differential equations, signal processing, and viscoelastic models.