On the Analyticity of Laguerre Series

Abstract

The transformation of a Laguerre series $f (z) = \sum_{n=0}^{\infty} λ_{n}^{(α)} L_{n}^{(α)} (z)$ to a power series $f (z) = \sum_{n=0}^{\infty} γ_{n} z^{n}$ 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 $λ_{n}^{(α)}$ as $n \to \infty$ can be formulated which guarantee that $f (z$ is analytic at $z=0$. Meaningful result are obtained if the $λ_{n}^{(α)}$ either decay exponentially or factorially as $n \to \infty$. The situation is much more complicated if the $λ_{n}^{(α)}$ decay algebraically as $n \to \infty$. If the $λ_{n}^{(α)}$ 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 $λ_{n}^{(α)}$ 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 $λ_{n}^{(α)}$, 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 $λ_{n}^{(α)}$ 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.