New properties and representations for members of the power-variance family.I

We derive new Wright-function representations for the densities of the generating measures of most representatives of the power-variance family of distributions. For all members of this family, we construct new saddlepoint-type approximations having an arbitrary fixed number of refining terms. To this end, we derive new, “exponentially small,” Poincaré series for a subclass of the Wright functions whose coefficients are expressed in terms of the Zolotarev polynomials. MSC: primary 60E07, 60E10, 60F10, 62E10, 62E20; secondary 33C10, 33C15, 33E20, 41A60