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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 $\vert z\vert<1$, the authors propose a new method to approximate the late coefficients $a_n$ of the Maclaurin expansion $F(z)=\sum_{n=0}^\infty a_nz^n$ when $F(z)$ has only a finite number of algebraic singularities on the unit circle $\vert z\vert=1$. 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 $F(z)$ on the unit circle $\vert z\vert=1$ 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 $P_n^{(\lambda)}(x)$. Updated 2.2.2006

41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
33E20Functions defined by series and integrals
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