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

Vladimir Vinogradov; Richard B. Paris; Olga Yanushkevichiene

doi:10.1007/s10986-012-9186-0

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

Vladimir Vinogradov, Richard B. Paris, Olga Yanushkevichiene

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Abstract

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 language	English
Pages (from-to)	444–461
Number of pages	18
Journal	Lithuanian Mathematical Journal
Volume	52
Issue number	4
DOIs	https://doi.org/10.1007/s10986-012-9186-0
Publication status	Published - Oct 2012

Keywords

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

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title = "New properties and representations for members of the power-variance family.I",

abstract = "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,” Poincar{\'e} 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",

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TY - JOUR

T1 - New properties and representations for members of the power-variance family.I

AU - Vinogradov, Vladimir

AU - Paris, Richard B.

AU - Yanushkevichiene, Olga

PY - 2012/10

Y1 - 2012/10

N2 - 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

AB - 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

U2 - 10.1007/s10986-012-9186-0

DO - 10.1007/s10986-012-9186-0

M3 - Article

VL - 52

SP - 444

EP - 461

JO - Lithuanian Mathematical Journal

JF - Lithuanian Mathematical Journal

SN - 0363-1672

IS - 4

ER -

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

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Keywords

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