On a Systematic Approach to some Properties of Special Functions

It can be shown that several important special functions, e.g. Gaussian and confluent hypergeometric, associated Legendre and Laguerre, and Gegenbauer and Jacobi types, can be expressed as derivatives of complex order of elementary functions. These expressions lead, through straightforward application of the rules of differentiation with complex order, to short, one- or two-line proofs of recurrence, contiguity, and derivative formulas for these special functions; if the definition of derivative of complex order is used, the same expressions readily yield representations of these special functions, as integrals along the real axis or paths in the complex plane. It is shown that the definition of derivative of complex order leads to integral identities, which allow the integrals of products of special and elementary functions to be evaluated as a single function; also, derivatives of complex order satisfy series identities, which sum products of special functions as a single special function, and thus generalize some series of classical orthogonal polynomials to the corresponding special functions with complex parameters. In conclusion, it is demonstrated that the combination of the rules of differentiation with complex order with the definition of special functions as derivatives of complex order of elementary functions yields a simple method of systematic derivation of a number of important properties of widely used special functions. 1