Exponentially small expansions of the Wright function on the Stokes lines

Richard B. Paris

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Abstract

We investigate a particular aspect of the asymptotic expansion of the Wright function pΨq(z) for large |z|. In the case p = 1, q ⩾ 0, we establish the form of the exponentially small expansion of this function on certain rays in the z-plane (known as Stokes lines). The importance of such exponentially small terms is encountered in analytic probability theory and in the theory of generalised linear models. In addition, the transition of the Stokes multiplier connected with the subdominant exponential expansion across the Stokes lines is shown to obey the familiar error-function smoothing law expressed in terms of an appropriately scaled variable. Some numerical examples which confirm the accuracy of the expansion are given.
Original languageEnglish
Pages (from-to)82–105
Number of pages24
JournalLithuanian Mathematical Journal
Volume54
Issue number1
DOIs
Publication statusPublished - Jan 2014

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Wright Function
Stokes
Line
Error function
Generalized Linear Model
Probability Theory
Multiplier
Asymptotic Expansion
Half line
Smoothing
Numerical Examples
Term

Cite this

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Exponentially small expansions of the Wright function on the Stokes lines. / Paris, Richard B.

In: Lithuanian Mathematical Journal, Vol. 54, No. 1, 01.2014, p. 82–105.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Exponentially small expansions of the Wright function on the Stokes lines

AU - Paris, Richard B.

PY - 2014/1

Y1 - 2014/1

N2 - We investigate a particular aspect of the asymptotic expansion of the Wright function pΨq(z) for large |z|. In the case p = 1, q ⩾ 0, we establish the form of the exponentially small expansion of this function on certain rays in the z-plane (known as Stokes lines). The importance of such exponentially small terms is encountered in analytic probability theory and in the theory of generalised linear models. In addition, the transition of the Stokes multiplier connected with the subdominant exponential expansion across the Stokes lines is shown to obey the familiar error-function smoothing law expressed in terms of an appropriately scaled variable. Some numerical examples which confirm the accuracy of the expansion are given.

AB - We investigate a particular aspect of the asymptotic expansion of the Wright function pΨq(z) for large |z|. In the case p = 1, q ⩾ 0, we establish the form of the exponentially small expansion of this function on certain rays in the z-plane (known as Stokes lines). The importance of such exponentially small terms is encountered in analytic probability theory and in the theory of generalised linear models. In addition, the transition of the Stokes multiplier connected with the subdominant exponential expansion across the Stokes lines is shown to obey the familiar error-function smoothing law expressed in terms of an appropriately scaled variable. Some numerical examples which confirm the accuracy of the expansion are given.

U2 - 10.1007/s10986-014-9229-9

DO - 10.1007/s10986-014-9229-9

M3 - Article

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SP - 82

EP - 105

JO - Lithuanian Mathematical Journal

JF - Lithuanian Mathematical Journal

SN - 0363-1672

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