A Steepest Descent Method for the Asymptotic Solutions 1 of a Second-Order Difference Equation 337 最陡下法推導二階差分方程式漸進解最陡下法推導二階差分方程式漸進解 A Steepest Descent Method for the Asymptotic Solutions of a Second-Order Difference Equation 周章 Jang Jou 國際企業管系 吳鳳技術學院 Department of International Business Management WuFeng Institute of Technology 摘要：摘要：就一個由二廣義格爾函及一些基本函之乘積所構成的積分，考慮其所 滿足之二階差分方程式。此方程式之一對線性獨解乃係此差分方程式之最小解及主 勢解，且與合超幾何函 及有關。據此，本文用最陡下法推導出此方程 式之一對線性獨解分別的漸進公式。 關鍵詞：關鍵詞：最陡下法，漸進表示式，合超幾何函 Abstract : Using the method of steepest descent, asymptotic formulas are obtained for a pair of linearly independent solutions of a second-order difference equation which is satisfied by an integral involving the product of two generalized Laguerre polynomials with some elementary functions. The pair correspond to minimal and dominant solutions of the difference equation and are related to the confluent hypergoemetric functions and . Keywords s : Method of Steepest Descent, Asymptotic Representation, Confluent Hypergeometric Function. 1. Introduction Consider the second-order difference equation ), 3, 2, 1(011L=+nybyaynnnnn(1-1) where na and nb are a set of given real numbers with 0nb. The general solution of (1-1) can be represented as a linear combination of a pair of linearly independent solutions nf and ng. It is known 5 that if the two solutions are such that 0lim= nnngf(1-2) then serious problem of numerical instability will occur when one tries to compute the 2 吳鳳學報第 15 期 338 solutionnf, or any constant multiple of it by using the recurrence relation (1-1) in forward direction. A simple and effective way to remedy this problem is to apply the recurrence relation in backward direction. This is first introduced for the computation of modified Bessel functions )(xIn and is known as the Miller algorithm 3. Solutions such as nf are often called minimal solutions 6, while solutions such as ng which satisfy (1-2) with respect to a minimal solution nf are called dominant. For a review on this technique, see for example 7,9,11,12,19,20,22,23,25. When implementing the Miller algorithm to compute a minimal solution, one usually chooses to start with the initial values 0=nyand 01=ny for a sufficiently large n. These initial values do not correspond to the true values of the solution. Questions therefore naturally arise as to what values must be used for n and 1n to start the backward recurrence procedure in order that the generated sequence will converge to the true minimal solution with a prescribed accuracy. In the process of obtaining such an estimate on the initial values of n, it is often necessary to make use of asymptotic representations of the intended solutions for large n values. In this regard, a powerful and convenient technique can be found in the method of steepest descent originally devised by P. Debye, who won the Nobel prize in chemistry in 1936. Materials relevant to the method are often included in books with topics on complex contour integrals. See, for example, 4,8,13,14,17. Asymptotic formulas derived using this approximation technique generally may not rank in rigor and completeness with those established using the more sophisticated theory of approximation, but they nonetheless often prove adequate for the problems at hand, see for example 2,15,16. Besides, they are sometimes more readily available to practitioners in applied mathematics who do not possess the necessary expertise to benefit directly from mathematical literature. Handbooks on mathematical function, even one as comprehensive as 1, usually deal primarily with asymptotic behaviors for large values of coordinate variables but not for large values of the function order, at least for the case considered in this work. Above all, the functions which one considers may not be standard enough as to be found in any of published references. It is then imperative that a first-hand derivation of the required asymptotic solution formulas be carried out. In this paper, we are interested in obtaining asymptotic representations for both minimal and dominant solutions of the following second-order equation 012=+nnnnnnycybya (1-3) in which the coefficients are given by )2)(2(+=nnan (1-4) 232+= nbn (1-5) ) 1)(1(+=nncn . (1-6) Details on how (1-3) arises and a proof that the technique of backward recurrence is A Steepest Descent Method for the Asymptotic Solutions 3 of a Second-Order Difference Equation 339 applicable can be found in authors previous work 10. A prescription has also been given there for the initial values of n which will ensure solutions with any prescribed numerical accuracy. In Section 2, some key results bearing on the minimal and dominant solutions of (1-3) are recapitulated from 10. Integral