Abstract
Since 1970, we can find in the literature an important development concerning the fractional derivative theory. A large number of such familiar formulas from the elementary calculus have been shown to be special cases of more general expressions involving fractional derivatives. Taylor's and Laurent's series, the chain rule and Lagrange's expansion are such examples. In this paper, we add to this theory the following transformation formula for fractional derivatives: where α and p are arbitrary complex numbers. We explore many applications to special functions and several new summation formulas arising from the Darboux formula involving the classical orthogonal polynomials and Abel's identities are obtained.
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