Effective Laguerre Asymptotics

It is known that the generalized Laguerre polynomials can enjoy subexponential growth for large primary index. In particular, for certain fixed parameter pairs (a, z) one has the large-n asymptotic behavior L(−a) n (−z) ∼ C(a, z)n−a/2−1/4e2 √ . We introduce a computationally motivated contour integral that allows efficient numerical Laguerre evaluations, yet also leads to the complete asymptotic series over the full parameter domain of subexponential behavior. We present a fast algorithm for symbolic 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. A primary tool for these developments is an “exp-arc” method giving a natural bridge between converging series and effective asymptotics. 1 The challenge of “effectiveness” For primary integer indices n = 0, 1, 2, . . . we define the Laguerre polynomial thus: