Fractional Calculus of the Lerch Zeta Function

Abstract

This paper deals with the fractional derivative of the Lerch zeta function. We compute the fractional derivative of the Lerch zeta function using a complex generalization of the Grünwald–Letnikov derivative. This definition of fractional derivative, fulfilling the generalized Leibniz rule, allows us to derive a functional equation for the fractional derivative of the Lerch zeta function. This functional equation is rewritten in a simplified form that reduces its computational cost. Furthermore, we prove an approximate functional equation for the fractional derivative of the Lerch zeta function.