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.
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References
M.V. Berry, Uniform asymptotic smoothing of Stokes’s discontinuities, Proc. R. Soc. London, Ser. A, 122:7–21, 1989.
B.L.J. Braaksma, Asymptotic expansions and analytic continuations for a class of Barnes-integrals, Compos. Math., 15:239–341, 1963.
R.B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, London, 1973.
I.A. Ibragimov and Yu.V. Linnik, Independent and Stationary Sequences of Random Variables, Walters–Noordhoff, Groningen, 1971.
Yu.V. Linnik, On stable probability laws with exponent less than one, Dokl. Akad. Nauk SSSR, 94:619–621, 1954 (in Russian).
F.W.J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974. Reprinted by A.K. Peters, Wellesley, MA, 1997.
F.W.J. Olver, On Stokes’ phenomenon and converging factors, in R. Wong (Ed.), Proceedings of the International Conference on Asymptotic and Computational Analysis (Winnipeg, Canada, June 5–7, 1989), Marcel Dekker, New York, 1990, pp. 329–355.
F.W.J. Olver, Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral, SIAM J. Math. Anal., 22:1460–1474, 1991.
R.B. Paris, Smoothing of the Stokes phenomenon using Mellin–Barnes integrals, J. Comput. Appl. Math., 41:117–133, 1992.
R.B. Paris, Smoothing of the Stokes phenomenon for high-order differential equations, Proc. R. Soc. London, Ser. A, 436:165–186, 1992.
R.B. Paris, Exponential asymptotics of the Mittag-Leffler function, Proc. R. Soc. London, Ser. A, Math. Phys. Eng. Sci., 458:3041–3052, 2002.
R.B. Paris, Exponentially small expansions in the asymptotics of the Wright function, J. Comput. Appl. Math., 234:488–504, 2010.
R.B. Paris, Hadamard Expansions and Hyperasymptotic Evaluation, Cambridge Univ. Press, Cambridge, 2011.
R.B. Paris, Exponential smoothing of the Wright function, Technical Report MS 11:01, University of Abertay Dundee, 2011.
R.B. Paris and D. Kaminski, Asymptotics and Mellin–Barnes Integrals, Cambridge Univ. Press, Cambridge, 2001.
R.B. Paris and V. Vinogradov, Refined local approximations for members of some Poisson–Tweedie EDMs, 2013 (in preparation).
R.B. Paris and A.D. Wood, Asymptotics of High Order Differential Equations, Pitman Res. Notes Math. Ser., Vol. 129, Longman Scientific and Technical, Harlow, 1986.
A.V. Skorokhod, Asymptotic formulas for stable distribution laws, Dokl. Akad. Nauk SSSR, 98:731–734, 1954 (in Russian).
L.J. Slater, Generalized Hypergeometric Functions, Cambridge Univ. Press, Cambridge, 1966.
V. Vinogradov, R.B. Paris, and O. Yanushkevichiene, New properties and representations for members of the power-variance family. I, Lith. Math. J., 52:444–461, 2012.
V. Vinogradov, R.B. Paris, and O. Yanushkevichiene, New properties and representations for members of the power-variance family. II, Lith. Math. J., 53:103–120, 2013.
V. Vinogradov, R.B. Paris, and O. Yanushkevichiene, The Zolotarev polynomials revisited, in XXXI International Seminar on Stability Problems for Stochastic Models, Institute of Informatics Problems, Russian Academy of Sciences, Moscow, 2013, pp. 68–70.
R. Wong and Y.-Q. Zhao, Smoothing of Stokes’s discontinuity for the generalized Bessel function, Proc. R. Soc. London, Ser. A, Math. Phys. Eng. Sci., 455:1381–1400, 1999.
R. Wong and Y.-Q. Zhao, Exponential asymptotics of the Mittag-Leffler function, Constr. Approx., 18:355–385, 2002.
E.M. Wright, The asymptotic expansion of the generalized hypergeometric function, J. London Math. Soc., 10:286–293, 1935.
E.M. Wright, The asymptotic expansion of the generalized hypergeometric function, Proc. London Math. Soc. (2), 46:389–408, 1940.
V.M. Zolotarev, Expression of the density of a stable distribution with exponent α greater than one by means of a density with exponent 1/α, Dokl. Akad. Nauk SSSR, 98:735–738, 1954 (in Russian).
V.M. Zolotarev, One-Dimensional Stable Distributions, Amer. Math. Soc., Providence, RI, 1986.
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Paris, R.B. Exponentially small expansions of the Wright function on the Stokes lines. Lith Math J 54, 82–105 (2014). https://doi.org/10.1007/s10986-014-9229-9
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DOI: https://doi.org/10.1007/s10986-014-9229-9