On a uniform treatment of Darboux's method
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 \(F(z)=\sum_{n=0}^\infty a_nz^n\) 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 \(n\to\infty\). 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 \(P_n^{(\lambda)}(x)\). Updated 2.2.2006
scientific article; zbMATH DE number 2157507
scientific article