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

Vladimir Vinogradov, Richard B. Paris, Olga Yanushkevichiene

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    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
    Original languageEnglish
    Pages (from-to)444–461
    Number of pages18
    JournalLithuanian Mathematical Journal
    Issue number4
    Publication statusPublished - Oct 2012


    • Difference quotient
    • Poincaré series
    • Poisson-gamma laws
    • Reciprocity
    • Refined saddlepoint approximations
    • Stable laws
    • Stokes phenomenon
    • Wright function
    • Zolotarev duality


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