Department of Mathematics and Statistics, San Diego State University
Recently Andreas Strombergsson of Bristol University, UK, asked me about the asymptotics of the W Whittaker confluent hypergeometric function. This W function can be expressed in terms of the famous U confluent hypergeometric function. The desired asymptotic results are required for obtaining bounds on certain sums of the Fourier coefficients of Maass Wave eigenfunctions, and also for estimates on the equidistribution of non-closed horocycles on non-compact surfaces. The W function has two parameters and , along with a dependent variable x. The asymptotics he asked for are non-trivial! Namely, he wants purely imaginary with , uniformly for as well as for . I show how such results can be obtained using a general asymptotic theory from O.D.E.s that I obtained at SDSU in 1990. In this application it is necessary to take x into the complex plane. |