On the Analyticity of Laguerre Series

TL;DR

This paper investigates when Laguerre series can be transformed into power series that are analytic at zero, providing conditions based on decay rates and sign patterns of the coefficients, and discusses summation techniques for divergent series.

Contribution

It offers new sufficient conditions for the analyticity of functions represented by Laguerre series, especially for algebraically decaying coefficients, and explores summation methods for divergent series.

Findings

Exponential or factorial decay of coefficients guarantees analyticity.

Same sign coefficients lead to divergence and non-analyticity.

Alternating sign coefficients allow summation and analytic representation.

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. If the $\lambda_{n}^{(\alpha)}$ ultimately have strictly alternating signs, the series expansions for the power series coefficients are summable and the power series represents an analytic function. In the case of simple $\lambda_{n}^{(\alpha)}$, the summation of the divergent series for the power series coefficients can often be accomplished with the help of analytic continuation formulas for hypergeometric series, but if the $\lambda_{n}^{(\alpha)}$ 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.