On a Concept of Derivative of Complex Order with Applications to Special Functions

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Abstract
We introduce a concept of derivative Dν of complex order ν of a function f(z), generalizing both the Cauchy and Weyl integrals, which are the particular cases of, respectively, ν = +n/−n, a positive/negative integer, corresponding to ordinary derivation/integration of order n. Theorems of existence, analyticity and integrability are demonstrated for the derivative, of general complex order ν, including non-integer values, of functions f(z) either analytic or with one or more branch-points. Among the thirteen general properties (P1 to P13) proved for the derivative of complex order, are the null-theorems for the zero and non-zero constants, and non-triviality theorems for non-constant functions. The rules of association DμDν = Dμ + ν and commutation DμDν = DνDμ, for the derivatives of complex orders μ, ν, such that μ, ν, μ + ν are not negative integers, are proved; the case μ = −ν is used to introduce the operator inverse to the derivative of complex order ν, namely, the primitive Pν of complex order, which coincides with D−ν, the derivative of complex order −ν. As an application, thirteen representations (R1 to R13) of useful special functions are given, as derivatives of complex order of elementary functions; these 25 formulae generalize the Rodrigues-type formulae for the classical orthogonal polynomials to special functions, and apply with unrestricted complex values of all parameters. The results concern hypergeometric and confluent hypergeometric functions, Legendre, Laguerre and associated functions, Bessel and modified functions, and Hermite, Tchebycheff, Gegenbauer and Jacobi functions.

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