On the Analyticity of Laguerre Series

Abstract

The transformation of a Laguerre series $f(z)={\sum }_{n=0}^{\infty }{\lambda }_n^{(\alpha )}{L}_n^{(\alpha )}(z)$ to a power series $f(z)={\sum }_{n=0}^{\infty }{\gamma }_n{z}^n$ is discussed. Many nonanalytic functions can be expanded in this way. Thus, success is not guaranteed. Simple sufficient conditions based on the decay rates and sign patters of the ${\lambda }_n^{(\alpha )}$ as $n\to \infty$ can be formulated which guarantee that $f(z$ is analytic at $z=0$. Meaningful result are obtained if the ${\lambda }_n^{(\alpha )}$ either decay exponentially or factorially as $n\to \infty$. The situation is much more complicated if the ${\lambda }_n^{(\alpha )}$ decay algebraically as $n\to \infty$. If the ${\lambda }_n^{(\alpha )}$ ultimately have the same sign, the series expansions for the power series coefficients diverge, and the corresponding function is not analytic at $z=0.Ifthe$λ_{n}^{(α)}$ultimatelyhavestrictlyalternatingsigns,theseriesexpansionsforthepowerseriescoefficientsaresummableandthepowerseriesrepresentsananalyticfunction.Inthecaseofsimple$λ_{n}^{(α)}$,thesummationofthedivergentseriesforthepowerseriescoefficientscanoftenbeaccomplishedwiththehelpofanalyticcontinuationformulasforhypergeometricseries,butifthe$λ_{n}^{(α)}$ are more complicated, numerical techniques have to be employed. Certain nonlinear sequence transformations -- in particular the so-called delta transformation [E. J. Weniger, Comput. Phys. Rep. \textbf{10}, 189 -- 371 (1989), Eq. (8.4-4)] -- sum the divergent series occurring in this context effectively.

50 pages, LaTeX2e, 0 figures. To appear in Journal of Physics A