Laplace approximation of Lauricella functions F A and F D
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Laplace approximation of Lauricella functions FA and FD Ronald W. Butler · Andrew T. A. Wood
Received: 7 August 2013 / Accepted: 4 November 2014 © Springer Science+Business Media New York 2014
Abstract The Lauricella functions, which are generalizations of the Gauss hypergeometric function 2 F1 , arise naturally in many areas of mathematics and statistics. So far as we are aware, there is little or nothing in the literature on how to calculate numerical approximations for these functions outside those cases in which a simple one-dimensional integral representation or a one-dimensional series representation is available. In this paper we present first-order and second-order Laplace approximations to the Lauricella functions FA(n) and FD(n). Our extensive numerical results show that these approximations achieve surprisingly good accuracy in a wide variety of examples, including cases well outside the asymptotic framework within which the approximations were derived. Moreover, it turns out that the second-order Laplace approximations are usually more accurate than their first-order versions. The numerical results are complemented by theoretical investigations which suggest that the approximations have good relative error properties outside the asymptotic regimes within which they were derived, including in certain cases where the dimension n goes to infinity. Keywords Gauss hypergeometric function · Lauricella functions · Vector-argument hypergeometric functions
Communicated by: Alexander Barnett R. W. Butler Statistical Science Department, Southern Methodist University, Dallas, TX 75275-0332 USA A. T. A. Wood () School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK e-mail: [email protected]
R.W. Butler, A.T.A. Wood
Mathematics Subject Classifications (2010) 33 · 41 · 62
1 Introduction (n)
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The Lauricella functions FA , FB , FC and FD were introduced in the case n = 3 by Lauricella [10]. Each of these functions is a generalization of the classical Gauss hypergeometric function 2 F1 (e.g. Abramowitz and Stegun, [1], Chapter 15), and 2 F1 is recovered when n = 1. An extensive account of many of the mathematical properties of Lauricella functions for a general positive integer n, and discussion of problems in mathematics and statistics in which they arise, are given in the book by Exton [7]. These functions appear in a wide variety of settings; see, for instance, Dickey [6]; Lijoi and Regazzini [11]; Kerov and Tsilevich [9] and Scarpello and Ritelli [15]. Our starting point in this paper is the question of how to calculate good approximations for Lauricella functions, a problem which, so far as we are aware, has received little or no attention in the literature. We derive first- and second-order (n) (n) Laplace approximations for the functions FA and FD , focusing on those situations in which convenient one-dimensional integral representations or one-dimensional series representations are not available. Our numerical results indicate that it is nearly always
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