Uniform asymptotic expansions involving exponential and Airy functions areobtained for Laguerre polynomials $L_{n}^{(\alpha)}(x)$, as well ascomplementary confluent hypergeometric functions. The expansions are valid for$n$ large and $\alpha$ small or large, uniformly for unbounded real and complexvalues of $x$. The new expansions extend the range of computability of$L_n^{(\alpha)}(x)$ compared to previous expansions, in particular with respectto higher terms and large values of $\alpha$. Numerical evidence of theiraccuracy for real and complex values of $x$ is provided.
展开▼
机译：对于Laguerre多项式$ L_ {n} ^ {（\ alpha）}（x）$以及互补的合流超几何函数，可以获得涉及指数和Airy函数的一致渐近展开。扩展对于$ n $大和$ \ alpha $小或大有效，统一适用于$ x $的无穷实数值和复数值。与先前的扩展相比，新的扩展扩展了$ L_n ^ {（\ alpha}}（x）$的可计算性范围，尤其是在$α_alpha$的较高项和较大值方面。提供了其对$ x $的实际和复杂值的准确性的数字证据。
展开▼
