Whittaker functions with one or both parameters large: simplified uniform asymptotic expansions involving Bessel and Airy functions

Abstract

Uniform asymptotic expansions are derived as $\kappa \to \infty$ for the Whittaker functions $W_{\kappa ,\mu }(z)$, $M_{\kappa ,\mu }(z)$, as well as related functions including generalized Laguerre polynomials. The results are uniformly valid for $0\le \mu \le (1-{\delta }_0)\kappa <\kappa$, where ${\delta }_0\in (0,1)$ 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 $\mu$.