Exponentially small expansions of the Wright function on the Stokes lines
- 105 Downloads
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.
Keywordsasymptotics exponentially small expansions Wright function Stokes lines
MSC33C20 33C70 34E05 41A60
Unable to display preview. Download preview PDF.
- 6.F.W.J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974. Reprinted by A.K. Peters, Wellesley, MA, 1997.Google Scholar
- 7.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.Google Scholar
- 14.R.B. Paris, Exponential smoothing of the Wright function, Technical Report MS 11:01, University of Abertay Dundee, 2011.Google Scholar
- 16.R.B. Paris and V. Vinogradov, Refined local approximations for members of some Poisson–Tweedie EDMs, 2013 (in preparation).Google Scholar
- 17.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.Google Scholar
- 22.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.Google Scholar
- 25.E.M. Wright, The asymptotic expansion of the generalized hypergeometric function, J. London Math. Soc., 10:286–293, 1935.Google Scholar