Abstract: A new version of the Taylor series expansion is obtained by ascribing a typical value of complex variable in the expansion. Applications to the usual special functions, including hypergeometric function, spherical harmonics function, cylindrical function, and modified cylindrical function are made. About 20 identities are gained for these functions, which can not be found in the classical literatures.

Key words: Taylor series expansion, hypergeometric functions, spherical functions, cylindrical funct