# Effective Laguerre asymptotics

Borwein, David and Borwein, Jonathan M. and Crandall, Richard E. (2008) Effective Laguerre asymptotics. SIAM J. Numerical Anal., 6 . pp. 3285-3312.

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It is known that the generalized Laguerre polynomials can enjoy sub-exponential growth for large primary index. Specifically, for certain fixed parameter pairs $(a,z)$ one has the large-$n$ asymptotic $$L_n^{(-a)}(-z) \sim C(a,z) n^{-a/2-1/4} e^{2\sqrt {nz}}.$$ We introduce a computationally motivated contour integral that allows highly efficient numerical evaluation of $L_n$, yet also leads to general asymptotic series over the full domain for sub-exponential behavior. We eventually lay out a fast algorithm for generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i.e. rigorous and explicit) error bounds on the general expansion. To this end, we avoid classical stationary-phase and steepest-descent techniques in favor of an exp-arc" method that amounts to a natural bridge between converging series and effective asymptotics. Finally, we exhibit an absolutely convergent exp-arc series for Bessel-function evaluation as an alternative to conventional ascending-asymptotic switching.