Mathematics – Classical Analysis and ODEs
Scientific paper
2008-08-28
Mathematics
Classical Analysis and ODEs
50 pages, LaTeX2e, 0 figures. To appear in Journal of Physics A
Scientific paper
10.1088/1751-8113/41/42/425207
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.
No associations
LandOfFree
On the Analyticity of Laguerre Series does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the Analyticity of Laguerre Series, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the Analyticity of Laguerre Series will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-327503