2008/01/01 by David Borwein, Jonathan M. Borwein, Richard E. Crandall · 1 citation
Mathematics · #Advanced Mathematical Identities #Fractional Differential Equations Solutions #Mathematical functions and polynomials
paper · doi:10.1137/07068031x
openalex publication_date 2008/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23
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 Ln(-a)(-z) ∼ C(a,z) n-a/2-1/4 e2√(nz). 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. (A corrected version of this paper has been appended to the originally posted pdf).