Integral representations of one-dimensional projections for multivariate stable densities

We consider the numerical evaluation of one-dimensional projections of general multivariate stable densities introduced by Abdul-Hamid and Nolan [H. Abdul-Hamid, J.P. Nolan, Multivariate stable densities as functions of one dimensional projections, J. Multivariate Anal. 67 (1998) 80–89]. In their approach higher order derivatives of one-dimensional densities are used, which seems to be cumbersome in practice. Furthermore there are some difficulties for even dimensions. In order to overcome these difficulties we obtain the explicit finite-interval integral representation of one-dimensional projections for all dimensions. For this purpose we utilize the imaginary part of complex integration, whose real part corresponds to the derivative of the one-dimensional inversion formula. We also give summaries on relations between various parametrizations of stable multivariate density and its one-dimensional projection.