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Uniform asymptotic expansion of the Jacobi polynomials in a complex domain. (English) Zbl 1078.33008
The paper studies two related problems. The first problem is to find an asymptotic formula of Jacobi polynomials that includes both the exponential and oscillatory approximations. This problem is analogous to the problem associated with the turning-point theory for differential equations. The second problem is to demonstrate that the equation proposed actually provides an exponentially improved asymptotic expansion for the Jacobi polynomials. An asymptotic formula is found that links the behaviour of the Jacobi polynomial in the interval of orthogonality $\left[-1,1\right]$ with that outside the interval. This formula holds uniformly in the exterior of an arbitrary closed curve which encloses the interval $\left[-1,1\right]$ in the sense that the ratio tends uniformly to 1. The two infinite series involved in this formula are shown to be exponentially improved asymptotic expansions. The method used in this paper can also be adopted in other cases of orthogonal polynomials such as Hermite and Laguerre. Although in most cases asymptotic methods for differential equations give stronger results than the corresponding ones for integrals, the result in this paper presents one example in which the integral approach has an advantage over the differential-equation theory.
##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)