On Fractional derivative of Hurwitz Zeta function and Jacobi Theta function

Abstract

This research paper focuses on exploring two Complex-valued function's fractional derivative, specifically the Hurwitz Zeta function and Jacobi theta function. The study is based on the Complex Generalization of Grunwald-Letnikov Fractional derivative which adheres to the generalized version of the Leibniz rule. Within this paper, we present and establish an identity that serves as a Generalization of the Hurwitz Zeta function's Functional Equation. Additionally, we derive the Jacobi Theta function's Functional Equation, accompanied by a comprehensive examination of the properties associated with the Fractional derivative of these two functions.