AI 中文总结
该研究针对$\kappa \to \infty$时的惠特克函数$W_{\kappa,\mu}(z)$、$M_{\kappa,\mu}(z)$及相关函数，基于关联微分方程推导了涉及贝塞尔函数与艾里函数的简化一致渐近展开式，可覆盖复$z$平面主分支并可扩展至所有分支及负$\mu$。
AI 中文摘要
针对κ→∞时的惠特克函数W_{κ,μ}(z)、M_{κ,μ}(z)以及包括广义拉盖尔多项式在内的相关函数，推导得到了一致渐近展开式。该结果在0≤μ≤(1−δ₀)κ<κ范围内一致有效，其中δ₀∈(0,1)为任意固定值。分析基于具有一个二重极点和两个转折点的关联微分方程。在其中一个可与二重极点合并的转折点处，应用近期发展的渐近理论得到了涉及贝塞尔函数的展开式；在第二个转折点处，得到了涉及艾里函数的展开式。与早期结果中的系数不同，这两种情况的系数均可便捷计算。这些展开式组合后可在复z平面的主分支上一致覆盖整个平面，利用标准的解析延拓和连接公式可将结果扩展至z的所有分支以及负μ。
英文摘要
Uniform asymptotic expansions are derived as $κ\to \infty$ for the Whittaker functions $W_{κ,μ}(z)$, $M_{κ,μ}(z)$, as well as related functions including generalized Laguerre polynomials. The results are uniformly valid for $0\leqμ\leq(1-δ_0)κ<κ$, where $δ_0\in(0,1)$ is arbitrary and fixed. The analysis is based on the associated differential equation, which has a double pole and two turning points. At one of the turning points, which may coalesce with the double pole, a recently developed asymptotic theory is applied to obtain expansions involving Bessel functions. At the second turning point, expansions involving Airy functions are obtained. In both cases, the coefficients are readily computable, in contrast to those occurring in earlier results. The expansions, when taken together, uniformly cover the entire complex $z$-plane on the principal branch. Standard analytic continuation and connection formulas extend the results to all branches of $z$ and to negative $μ$.
