Taylor’s Series Generalized for Fractional Derivatives and Applications

Abstract

The familiar Taylor’s series expansion of the function , $f(z)$ has for its general term $D^n f(z_0 ){{(z - z_0 )^n } / {n!}}$. A new generalization of Taylor’s series in which the general term is $D^{an + \gamma } f(z_0 ){{(z - z_0 )^{an + \gamma } } / {\Gamma (an + \gamma + 1)}}$, where $a > 0$ and $\gamma $ is an arbitrary complex number, is examined. This new series is extended further to a form which includes the familiar Lagrange’s expansion as a special case. The derivatives appearing in this series are of order $an + \gamma $ and are called “fractional derivatives.” Examples of the use of this new series for discovering generating functions are given.