On generalizations of the series of Taylor, Lagrange, Laurent and Teixeira (Q1174749)

In this paper an extension is given of the classical theorems of Taylor, Lagrange, Laurent and Teixeira. The function \(F(z)\), which is analytic or has an isolated singularity at a point \(z=a\), is replaced by its derivative of complex order \(F^{(\nu)}(z)\) in a sense. The series representations of \(F^{(\nu)}(z)\) are involved with non-integral powers coming from fractional derivatives or integrals. As an application, generating functions are established for the Hermite and Bessel functions with complex parameters.