Taylor’s Series Generalized for Fractional Derivatives and Applications

The familiar Taylor’s series expansion of the function , $f(z)$ has for its general term Dⁿ f(z₀ )(z - z₀ )ⁿ / n!. A new generalization of Taylor’s series in which the general term is Dan + γ f(z₀ )(z - z₀ )an + γ / Γ (an + γ + 1), where $a > 0$ and γ 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 + γ and are called “fractional derivatives.” Examples of the use of this new series for discovering generating functions are given.