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

Given an analytic function $F\left(z\right)$ inside the unit circle $|z|<1$, the authors propose a new method to approximate the late coefficients ${a}_{n}$ of the Maclaurin expansion $F\left(z\right)={\sum }_{n=0}^{\infty }{a}_{n}{z}^{n}$ when $F\left(z\right)$ 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 $n\to \infty$. But this method does not work when the singularities of $F\left(z\right)$ 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 ${P}_{n}^{\left(\lambda \right)}\left(x\right)$.

Updated 2.2.2006

##### MSC:
 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 33E20 Functions defined by series and integrals