The Wright Function – Numerical Approximation and Hypergeometric Representation

The Wright function, a special function, which arises in the theory of the space-time fractional diffusion equation, is a very general mathematical object with diverse connections to other special and elementary functions. The paper presents a numerical integration technique for approximation of the Wright function \(W(a, b\, | x)\) using the method of stationary phase. As a second contribution, the paper exhibits a symbolic algorithm for hypergeometric (HG) representation of the Wright function. For rational values of its first argument the function is reduced to a finite sum of HG functions and polynomials. The HG functions are themselves represented by known elementary or other special functions, wherever possible. Reference implementations of the algorithms are programmed in the open-source computer algebra system Maxima.