Uniform asymptotic expansions for Laguerre polynomials and related confluent hypergeometric functions

Abstract

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

Submitted to Advances in Computational Mathematics