Derivatives of Mittag-Leffler Functions with Respect to their Parameters

The Mittag-Leffler functions are natural extensions of the exponential function and appear often as solutions of differential equations of non-integer order in much the same way as exponential functions appear as solutions of differential equations of integer order. This ubiquitous nature of Mittag-Leffler functions underscores the importance of understanding the properties of these functions. In this regard, the derivatives of the Mittag-Leffler function E$_{\alpha ,\beta }$(-x) with respect to its parameters $\alpha $ and $\beta $ have been investigated. Particularly interesting are the derivatives of t$^{\alpha -1}$E$_{\alpha ,\alpha }$(-t$^{\alpha })$, which occurs as the fundamental Green's function solution to certain dynamic problems.