Uniform asymptotic expansions for Laguerre polynomials and related confluent hypergeometric functions

TL;DR

This paper derives uniform asymptotic expansions for Laguerre polynomials and related confluent hypergeometric functions, enhancing their computational range for large degrees and various parameter values, with numerical validation.

Contribution

It introduces new uniform asymptotic expansions involving exponential and Airy functions for Laguerre polynomials and confluent hypergeometric functions, extending their computability.

Findings

Expansions valid for large n and varying alpha

Extensions to unbounded complex x values

Numerical evidence confirms high accuracy

Abstract

Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials $L_{n}^{(\alpha)}(x)$, as well as complementary confluent hypergeometric functions. The expansions are valid for $n$ large and $\alpha$ small or large, uniformly for unbounded real and complex values of $x$. The new expansions extend the range of computability of $L_n^{(\alpha)}(x)$ compared to previous expansions, in particular with respect to higher terms and large values of $\alpha$. Numerical evidence of their accuracy for real and complex values of $x$ is provided.