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On a uniform treatment of Darboux’s method. (English) Zbl 1074.41505

Given an analytic function F(z) inside the unit circle |z|<1, the authors propose a new method to approximate the late coefficients a n of the Maclaurin expansion F(z)= n=0 a n z n when F(z) has only a finite number of algebraic singularities on the unit circle |z|=1. It is well known that the Darboux method gives an asymptotic expansion of a n when n. But this method does not work when the singularities of F(z) on the unit circle |z|=1 approach each other, that is, the method is not uniform in the parameter which controls the confluence of singularities. A uniform method is known since 1968 for a special case of confluence, although it is too complicated. The authors propose in this paper a more simple uniform method somehow inspired in the previous work of Chester, Friedman, Ursell, Blestein, Olde Daalhuis and Temme. In the case of two coalescing singularities, the authors obtain a uniform expansion of a n for large n in terms of two sequences of inverse powers of n, each sequence multiplied by a Bessel functions of the first kind. In the general case of p coalescing singularities, the uniform expansion of a n is given in terms of p sequences of inverse powers of n, each sequence multiplied by more complicated functions than the Bessel functions. The method is applied to the ultraspherical polynomials P n (λ) (x).

Updated 2.2.2006

MSC:
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
33C10Bessel and Airy functions, cylinder functions, 0 F 1
33E20Functions defined by series and integrals